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University  of  California  •  Berkeley 

The  Theodore  P.  Hill  Collection 
Early  American  Mathematics  Books 


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NEW 


UNIVERSITY  ARITHMETIC, 


EMBRACING    THE 


SCIEJ^CE    OF    I^^UMBERS, 


AND    TUEIR 


APPLICATIONS  ACCORDIXG  TO  TIIK  MOST  I3IPKOVED  METHODS 


OF 


ANALYSIS    AND    CANCELLATION. 


BY 


CHARLES  DA  VIES,  LL.  D., 


AUTHOR    OF     rnniAKT,    INTELLECTUAL,    AND    SCHOOL   ARITHMETICS  ;    ELEMENTART 
ALQEBUA  ;    ELEMENTARY     GEOMETKY  ;    PRACTICAL     MATHEMATICS  :    ELEMENTS 
OF     SURVEYING  ;    ELEMENTS   OF    ANALYTICAL   GEOMETRY  ;    DESCRIPTIVE 
GEOMETRY  ;    f-UADES,    SHADOWS,    AND   PERSPECTIVE  ;    DIFFER- 
ENTIAL  AND   INTEGRAL   CALCULUS  ;     AND    LOGIC   AND 
UTILITY    OF  MATHEMATICS. 


NEW    YOrvK: 

PUBLISHED    BY    A  .    S.    BARNES    &    CO., 
No.    51   &  53    .1  O  I!  \  -  L^  T  11  E  E  T  . 
1858. 


Entered  according  to  act  of  Congress,  in  the  year  eighteen  hundred  and  fifty-six, 

BY    CnATtLES    DAVIES, 
In  the  Clerk's  OfRce  of  the  District  Court  of  the  United  States,  for  the  Southern 

District  of  New  York. 


JONES      &      DENYSE, 
STEREOTYPErtS    AND    ELECTROTYPERS, 
1S3  AViUiam-Strect. 


PEEFACE. 


Science,  in  its  popular  signification,  means  knoAA'ledga 
reduced  to  order ;  that  is,  knowledge  so  classified  and  arranged, 
as  to  he  easily  remembered,  readily  referred  to,  and  advanta- 
geously applied. 

Arithmetic  is  the  science  of  numbers.  It  is  the  foundation 
of  the  exact  and  mixed  sciences  and  a  knowledge  of  it  is  an 
important  element  either  of  a  liberal  or  practical  education. 
While  Arithmetic  is  a  science  in  all  that  concerns  the  properties 
of  numbers,  it  is  an  art  in  all  that  relates  to  their  practical 
application. 

It  is  the  first  subject  in  a  well-arranged  course  of  instruction 
to  which  the  reasoning  powers  of  the  mind  are  applied,  and  i? 
the  guide-book  of  the  mechanic  and  man  of  business.  It  is  the 
first  fountain  at  which  the  young  votary  of  knowledge  drinkf 
the  pure  waters  of  intellectual  truth. 

It  has  seemed,  to  the  author,  of  the  first  importance  that  this 
subject  should  be  well  treated  in  our  Elementary  Text  Books. 
In  the  hope  of  contributing  something  to  so  desirable  an  end,  he 
has  prepared  a  series  of  arithmetical  works,  embracing  luur 
books  enthled,  Primary  Arithmetic  ;  Intellectual  Arithmetics 
School  Arithmetic;  and  University  Arithmetic — the  latter  of 
which  is  the  present  volume. 

PuiMAny  AraTKMETic.  This  first  book  is  adapted  to  the 
capacities  and  wants  of  young  children.  Sensible  objects  are 
employed  to  illustrate  and  make  familiar  the  simple  combi- 
nations and  relations  of  numbers.  Each  lesson  embraces  one 
combination  of  numbers,  or  one  set  of  combinations. 


VI  PEEFACE. 

Intellectual  Arithmetic.  This  work  is  designed  to  pro- 
sent  a  thorough  analysis  of  the  science  of  numbers,  and  to  form 
a  complete  course  of  mental  arithmetic.  It  is  thought  to  be 
accessible  to  young  pupils  by  the  simplicity  and  gradation  of 
its  methods,  and  to  be  particularly  adapted  to  the  wants  of 
advanced  students,  as  the  attempt  has  been  faithfully  made  to 
give  the  subjects  of  which  it  treats  a  scientific  arrangement 
and  logical  connection  in  all  the  higher  methods  of  arithmeti- 
cal analysis. 

School  Arithmetic.  Great  pains  has  been  taken  in  the 
preparation  of  this  book  to  combine  theory  and  practice  ;  to 
explain  and  illustrate  principles,  and  to  apply  them  to  the 
common  business  transactions  of  life — to  make  it  einjyhatically 
a  2y''(^ctical  work.  The  student  is  required  to  demonstrate 
every  principle  laid  down,  by  a  course  of  mental  reasoning, 
before  deducing  a  proposition  or  making  a  practical  application 
of  a  rule  to  examples.  He  is  required  to  fix  upon  the  U7tit  or 
unity  as  the  bane  of  all  numbers,  whether  integral  or  frac- 
tional— to  reason  with  constant  reference  to  this  base,  and  thus 
make  it  the  key  to  the  solution  of  all  arithmetical  questions. 
It  is  thought,  that  the  language  used  in  the  statement  of  princi- 
ples, in  the  definitions  of  terms,  and  in  the  explanation  of 
methods,  will  be  ibund  to  be  clear,  exact,  brief  and  compre- 
hensive. 

Univeksity  Auitiimetic.  This  work  is  designed  to  answer 
another  object.  Here,  the  entire  subject  is  treated  as  a 
science.  The  scholar  is  supposed  to  be  familiar  with  the  simple 
operations  in  the  four  ground  rules,  and  with  the  first  principles 
of  fractions,  these  being  now  taught  to  small  children  either 
orally  or  from  elementary  treatises.  This  being  premised,  the 
language  of  figures,  which  are  the  representatives  of  numbers, 
is  carefully  taught,  and  the  different  .significations  of  which  the 
figures  are  susceptible,  depending  on  the  manner  in  whicli  they 
are  written,  are  fully  explained.  It  is  shown,  for  example, 
that    the    simple   numbers   in    which    the    value   of    the    unit 


PREFACE.  Vll 

increases  from  right  to  left  according  to  the  scale  of  tens,  and 
the  Denominate  or  Compound  numbers  in  which  it  increases 
according  to  a  varying  scale,  belong  to  the  same  class  of  num- 
bers, and  that  both  may  be  treated  under  the  same  rules. 
Hence,  the  rules  for  Notation,  Addition,  Subtraction,  Multipli- 
cation and  Division,  have  been  so  constructed  as  to  apply 
equally  to  all  numbers.  This  arrangement,  which  the  author 
has  not  seen  elsewhere,  is  deemed  an  essential  improvement  in 
the  science  of  Arithmetic. 

Li  developing  the  properties  of  numbers,  from  their  elemen- 
tary to  their  highest  combinations,  great  labor  has  been 
bestowed  in  classification  and  arrangement.  It  has  been  a 
leading  object  to  present  the  entire  subject  of  arithmetic  as 
forming  a  series  of  ilepeiident  and  coyinccled  joropositions  :  so 
that  the  pupil,  while  acquiring  useful  and  practical  knowledge, 
may  at  the  same  time  be  introduced  to  those  beautiful  methoda 
of  reasoning,  v.'hich  science  alone  teaches. 

Great  care  has  been  taken  to  demonstrate  every  proposition — 
to  give  a  complete  analysis  of  all  the  methods  employed,  from 
the  simplest  to  the  most  difficult,  and  to  explain  fully,  the 
reason  of  every  rule.  A  full  analysis  of  the  science  of  Num- 
bers has  developed  but  one  law  ;  viz  ,  ihe  law  tchich  connects 
all  the  units  of  arithmetic  with  the  rinit  one,  and  which  points 
out  the  relations  of  these  loiits  to  each  other. 

In  the  Appendix,  which  treats  of  Units,  Weights  and  Mea- 
sures, &c.,  the  methods  of  determining  the  Arbitrary  Unit,  as 
well  as  the  general  law  which  prevails  in  the.  formation  of 
numbers,  are  fully  explained.  1  cannot  too  earnestly  recom- 
mend this  part  of  the  work  to  the  special  attention  of  Teachers 
and  pupils 

In  fine,  the  attention  of  teachers  is  especially  invited  to  this 
work,  because  general  methods  and  general  rules  are  employed 
to  abridge  the  common  arithmetical  processes,  and  to  give  to 
them  a  more  scieniific  and  practical  character.  In  the  present 
edition    the    matter  is  presented  in  a  new  form  ;  the  arrange- 


Vlll  PEEFACE. 

mcnt  of  the  subjects  is  more  natural  and  scientific  ;  the 
methods  have  been  carefully  considered  ;  the  illustrations 
abridged  and  simplified  ;  the  definitions  and  rules  thoroughly 
revised  and  corrected  ;  and  a  very  large  number  and  variety  of 
practical  examples  have  been  added.  The  subjects  of  Frac- 
tions, Proportion,  Interest,  Percentage,  Alligation,  Analysis,  and 
Weights  and  Measures,  present  many  new  and  valuable  fea- 
tures, which  are  not  found  in  other  works. 

A  Key  to  the  present  work  has  also  been  published  for  the 
use  of  such  Teachers  as  may  desire  it, — prepared  with  great 
care,  containing  not  only  the  answers  and  solutions  of  all  the 
examples,  but  a  full  and  comprehensive  analysis  of  the  more 
difficult  ones. 

The  author  has  great  pleasure  in  acknowledging  the  interest 
which  Teachers  have  manifested  in  the  success  of  his  labors  : 
they  have  suggested  many  improvements,  both  in  rules  and 
methods,  not  only  in  his  elementary,  but  also  in  his  advanced 
works.  The  school  room  is  the  tribunal,  and  the  intelligent 
and  practical  teacher  the  judge,  before  whom  all  text-books 
must  stand  or  fill. 

The  terms,  "  Cause  and  Effect,"  used  in  developing  the  sub- 
ject of  Pi'oportion,  were  supposed  by  the  author,  at  tlie  time  of 
their  introduction  into  his  works,  to  be  common  literary  property. 
But  he  has  since  been  informed,  that  Prof  H.  N.  Robinson, 
the  distinguished  author  of  a  full  course  of  Mathematics,  chiims 
to  have  been  the  first  who  made  a  practical  application  of  these 
terms,  in  a  tangible  form,  to  Proportion,  as  published  in  liis 
Aritlimetic  of  185-i.  I  have  great  pleasure,  tlierefore,  in  assifrn- 
ing  to  their  proper  author,  the  invention  of  so  ingenious  and 
practical  a  method,  and  of  acknowledging  the  courtesy  of  beinw 
permitted  to  continue  its  use  in  my  books,  with,  however,  as  I 
deem,  important  differences  in  the  definitions  and  development 
of  the  rule. 


FiSHKiLL  Landing.  ") 


My,  1858. 


CONTENTS. 


FIRST   FIVE    El'LES. 

PAGES. 

Definitions , 13 

Expressing  N^umbers 1<, 

Notation  and  Numeration 15-24 

Formation  and  Nature  of  Number? 24—26 

Scales 26 

United  States  Money 2Y-29 

Integral  Units 29 

Properties  of  the   9's 30 

Reduction 31-35 

Addition 35-44 

Subtraction 44—53 

Multiplication 63-67 

Division 67-83 

Practice .84*-85* 

Longitude  and  Time 84-86 

Applications  in   the    Four    Rules 86-95 

Properties  of  Numbers 95-97 

Divisibility  of  Numbers 97-99 

Greatest  Common  Divisor 99-102 

Least  Common  Multiple 102-104 

Cancellation 104-108 

COMMON    FRACTIONS. 

Definition  of,  and  First  Principles 108-1 11 

The  six  Kinds  of  Fractions 111-113 

Six  Propositions 113-117 

Reduction  of  Common  Fractions 117-124 

Addition  of  Common  Fractions 124—129 

Subtraction  of  Common  Fractions 129-133 

Multiplication  of  Common  Fractions 133-137 

1 


COINTENTS. 

PAGES 

Division  of  Common  Fractions 137-141 

Complex  Fractions l-il-142 

Applications  in  Fractions 142-141 

Duodecimals 144-150 


DECIMAL   FRACTIONS. 

Definition  of  Decimals,  &c 150-151 

Decimal  Numeration  Table — First  Principles,  &c 151-156 

Addition  of  Decimals 156-15S 

Subtraction  of  Decimals, 158-160 

Multiplication  of  Decimals 160-162 

Contractions  in  Multiplication 162-164 

Division  of  Decimals , 164-168 

Contractions  in  Division 168-170 

Reduction  of  Common  Fractions  to  Decimals 170-171 

Reduction  of  Denominate  Decimals 171-175 

Circulating  or  Repeating  Decimals — Definition  of,  &c 175-179 

Reduction  of  Circulating  Decimals 179-184 

Addition  of  Circulating  Decimals 184 

Subtraction  of  Circulating  Decimals 184-185 

Multiplication  of  Circulating  Decimals 185-186 

Division  of  Circulating  Decimals 186 


CONTINUED    FEACTIONS. 

Definitions  and  Principles 186-189 

RATIO    AND    PROPORTION. 

Ratio  Defined 189 

Proportion  Defined 190 

Simple  and  Compound  Ratio   102 

Simple  Proportion,  or  Rule  of  Three 193 

Cause  and  Ellect  198 

Inverse  Proportion 199-205 

Compound   Proportion 205-209 

Partnership 209-214 


OONTEKTS.  XI 


PERCENTAGE 

PAGES 

Percentage  Defined  and  Illustrated 214-219 

Simple  Interest 219-232 

Compound   Interest 282-235 

Discount 235-237 

Banking 237-240 

Bank  Discount 240-242 

Commission , 242-245 

Stocks  and  Brokerage 245-250 

Profit  and  Loss 250-255 

Insurance 255-257 

Life  Insurance 257-259 

APPLICATIONS. 

Endo-wments 259-261 

Annuities 261-263 

Assessing  Taxes 263-267 

Custom  House  Business 267-272 

Equation  of  Payments 272-275 

Alligation 275 

Alligation  Medial 275-276 

Alligation  Alternate 276-282 

Coins  and  Currencies 282-283 

Exchange 283-295 

General  Average 295-298 

Tonnage  of  Vessels 298-300 


PO'WEES   AND    ROOTS. 

Involution 300-301 

Evolution 301-302 

Extraction  of  Square  Root 302-31 1 

Cube  Root 311-317 


AEITHMETICAt   PROGRESSION. 

Definition  of,  <fec 317-319 

Different  Cases 31 9-322 

General  Example* 322-323 


sal  CONTENTS. 


GEOMETRICAL    PROGEESSION. 

PAGES 

Definition  of,  <fcc 323 

Cases 323-32G 

Examples 326 


AlfALTSIS. 


Analysis  and  Promiscuous  Examples S27-351 


MENSrKATION. 

Mensuration  of  Surfaces 351-357 

Mensuration  of  Volumes 357-362 

Gauging 362-366 

Mecliaiiical  Powei'S 366-374 

Questions  in  Katural  Philosophy 374-383 


APPENDIX 

Different  Kinds  of  Units 3S4-3S8 

United  States  Money 38S-390 

English    Money 39(1-393 

Linear  Measure 393-394 

Cloth  Measure 394-395 

Square  Measure 395-396 

Sui-ve3-oi'8'   Measure 396 

Cul.ic   Measure 396-393 

■\Viiie  Mt-asure 398 

Beer  Measure 393 

Dry  Measure 399 

Aviiirdu[)ois  Weight 400 

Troy   Weiglit 401 

Apotliecaries'    Weiglit 401 

Measure  of  Time 402-405 

Circular  Measure 406-406 


UNIVERSITY  ARITHMETIC. 


DEFINITIONS. 

1.  A  single  thing,  is  called  one  or  a  unit.  A  number  is  a 
unit,  or  a  collection  of  units. 

2.  A  single  thing  of  a  collection,  is  called  the  unit  or  hase  of 
the  collection.  The  primary  base  of  every  number  is  the  unit 
one. 

3.  SciKNCE  treats  of  the  properties  and  relations  of  things: 
Art  is  the  practical  application  of  the  principles  of  Science. 

4.  ARiTiniETic  treats  of  numbers.  It  is  a  science  when  it 
determines  the  properties  and  relations  of  numbers ;  and  an 
art,  when  it  applies  principles  of  science  to  practical  purposes. 

5.  A  Proposition  is  something  to  be  done,  or  demonstrated. 

6.  An  Analysis  is  an  examination  of  the  separate  parts  of  a 
proposition. 

7.  An  Operation,  in  Arithmetic,  is  the  act  of  doing  some- 
thing with  numbers.  The  number  obtained  by  an  operation  ia 
called  a  result,  or  answer. 

1.  What  is  a  single  thing  called  ?     What  is  a  number  1 

2.  What  is  a  single  thing  of  a  collection  called  ?  What  is  the  primary 
base  of  nvery  uiunber  1 

3    Of  what  does  science  treat  1     \A''hat  is  art  f 

4.  Of  what  does  Aritlimetic  treat  1    When  is  it  a  science '    When  an  art' 

5.  What  is  a  proposition  1 

6.  What  is  an  analysis  1 

7.  What  is  an  operation  1     What  is  the  number  obtained  called  ? 


14  NOTATIOIC. 

8.  A  Rule  is  a  direction  for  performing  an  operation,  and 
may  either  be  inferi'ed  from  an  analysis,  or  deduced  from  a 
demonstration. 

9.  There  are  five  fundamental  processes  of  Arithmetic: 
Notation  and  Numeration,  Addition,  Subtraction,  Multiplication, 
and  Division. 

EXPRESSING  NUMBERS. 

10.  There  are  three  methods  of  expressing  numbers  : 

1st.  By  words,  or  common  language  ; 

2d.    By  letters,  called  the  Roman  method ; 

3d.   By  figures,  called  the  Arabic  method. 

BY  WORDS. 

11.  A  single  thing  is  called  -  -  -  -  One. 
One      and  one  more       -  -  -  -  Two. 
Two      and  one  more       -  -  -  _  Tliree. 
Three  and  one  more       -  -  -  -  Four 
Four    and  one  more       -  -  -  -  Five. 
Five     and  one  more       -  -  -  _  Siix. 
Six       and  one  more       -  -  -  -  Seven. 
Seven  and  one  more       -  -  -  _  Fight. 


Eight    and  one  more       -         -         .         -         Nine. 
Nine     and  one  more       -         _         -         -         Ten. 


&c.  &c.  &;c. 

Each  of  the  words,  or  terms,  one,  ttco,  three,  four,  k,Q.,(kQ\-\oics 
how  many  units  are  taken.  These  terms  are  generally  called 
numbers ;  though,  in  fact,  they  are  but  the  names  of  num- 
bers. 

8.  What  is  a  rule?     How  may  it,  bp  deduced  1 

9.  How  many   fundamental  processes  are  there  in  Arithmetic  1     What 
are  they] 

10.  How  many  methods  arc  there  of  expressing  numbers  1  What  are 
they] 

11.  What  docs  each  of  the  words,  one,  iiru,  three,  &c.,  denote]  What 
are  these  words  generally  called  ]     What  are  they,  in  fact  ] 


NOTATION. 


15 


NOTATION. 

12.  Notation  is  the  method  of  expressing  numbers  cither 
by  letters  or  figures.  The  metliod  by  letters,  is  called  Roman 
Nutation  ;  the  method  by  figures  is  called  Arabic  Notation, 

ROMAN  NOTATION. 

13.  In  the  Eoman  Notation,  seven  capital  letters  are  used, 
viz. :  I,  stands  for  one  ;  V,  for  five  ;  X,  for  ten  ;  L,  for  fifty , 
C,  for  one  hundred;  D,  for  five  hundred ;  and  M,  for  07ie  thou- 
sand. All  other  numbers  are  expressed  by  combining  these 
letters  according  to  the  following 

ROMAN  TABLE. 


I.   -  - 

-     -     One. 

LXX.  - 

Seventy. 

II.  -  - 

-     -     Two. 

LXXX. 

Eighty. 

III.-  - 

-     -     Three. 

XC.      - 

Ninety. 

IV..     - 

-     -     Four. 

C.    -    - 

One  hundred. 

V.  -    - 

.     -     Five. 

CC-    - 

Two  hundred. 

VI.-    -    . 

■     -     Six. 

ccc.  . 

Three  hundred. 

VII.    -    - 

•     -     Seven. 

cccc- 

Four  hundred. 

VIII.  -    ■ 

■     -     Eight. 

D.   -    - 

Five  hundred. 

IX.      -    ■ 

•     -     Nine. 

DC.      - 

Six  hundi-ed. 

X.  -    -    . 

•     .     Ten. 

DCC.   - 

Seven  hundred. 

XX.    -     - 

-     Twenty. 

DCCC- 

Eight  hundred. 

XXX.      - 

-     Thirty. 

DCCCC 

Nine  hundred. 

XL.     -     . 

-     Forty. 

M.  -    - 

One  thousand. 

L.  -     -    - 

•     -     Fifty. 

MD.     - 

Fifteen  hundred. 

LX.    -    - 

•     -     Sixty. 

MM.    - 

Two  thousand. 

Note. — This  Notation  was  used  by  the  Romans  :  hence,  its  name. 
It  is  still  used  for  dates,  numbering  of  chapters,  pages,  &c. 

The  principles  of  this  Notation  are  these  : 

1 .  Every  time  a  letter  is  repeated,  the  number  which  it  denotes  is 
also  repeated. 


12.  What  is  Notation  ?     What  is  the  method  by  letters  called  1     What 
is  the  method  by  figures  called  1 

13.  How  many  letters  are  used  in  the  Roman  Notation  1     What  are 
♦■hey  ■?     What  does  each  stand  for  1 

Note. — What  are  the  three  principles  of  this  Notation  1 


16  NOTATION. 

2.  If  a  letter  denoting  a  less  number  is  written  on  the  right  of  one 
denoting  a  greater,  their  sum  will  express  the  number. 

3.  If  a  letter  denoting  a  less  number  is  written  on  the  left  of  one 
denoting  a  greater,  their  difference  will  express  the  number. 

EXAMPLES    IN   ROMAN   NOTATION. 

Express  the  following  numbers  in  the  Roman  Notation: 

1.  Sixteen. 

2.  Fourteen. 

3.  Eighteen. 

4.  Sixty-nine. 

5.  Seventy-eight. 

6.  One  hundred  and  fifteen. 

7.  Four  hundred  and  nine. 

8.  Seven  hundred  and  fifty-one. 

9.  One  thousand  and  sixty. 

10.  Two  thousand  and  ninety-one. 

11.  Five  hundred  and  sixty-nine. 

12.  Seven  hundred  and  forty-five. 

13.  Nine  hundred  and  sixty-one. 

14.  Six  hundred  and  ninety-nine. 

15.  Nine  hundred  and  fifty-seven. 

16.  One  thousand  two  hundred  and  six. 

17.  Four  hundred  and  ninety-five. 

18.  Seven  hundred  and  fiity-five. 

19.  Eighteen  hundred  and  forty-seven. 

20.  Two  thousand  five  hundred  and  twenty. 

ARABIC  NOTATION. 
14.  Arabic  Notation  is  the  method  of  expressing  numhera 
by  figures.     Ten  figures  are  used,  and  Ihoy  form  the  alphabet 
of  the  Arabic  Notation. 


14.  What  is  Arabic  Notation !  How  many  figurrs  are  used  ?  "VA'hat 
do  they  form  1  Name  the  figures  '  What  does  the  0  express  !  What 
are  the  other  figures  called  1 


NOTATION.  17 


They  are,     0  called  cipher,  or  Naught. 

.1  -  -  One. 

2  -  -  -  Two. 

3  -  -  -  Three. 

4  -  -  -  Four. 

5  -  -  -  Five. 

6  -  -  -  Six. 

7  -  -  -  Seven. 

8  -  -  -  Ei";ht. 


o 


9  -         -  Nine. 

The  cipher  0,  expresses  no  value.  It  is  used  to  denote  the 
ahscnce  of  a  thing.  The  nine  other  figures  are  called  5iV/«e/?- 
cant  Jjgurcs,  or  Digits. 

15.  We  have  no  sino;le  figure  for  the  number  ten.  We 
therefore  combine  the  figures  already  known.  This  we  do  by 
writing  0  on  the  right  hand  of  1,  thus : 

10,  Avhicli  is  read,  ten. 

This  10  is  equal  to  ten  of  tlie  units  expressed  by  1.  It  is, 
however,  but  a  single  ten,  and  may  be  regarded  as  a  unit,  the 
value  of  which  is  ten  times  as  great  as  the  unit  1.  It  is  called 
a  unit  of  the  second  order. 

16.  When  two  figures  are  written  by  the  side  of  each  other, 
the  one  on  the  right  is  in  the  flace  of  units,  and  the  other  in 
the  place  of  tens,  or  of  units  of  the  second  order.  Each  unit  of 
the  second  order  is  equal  to  ten  units  of  the  first  order. 

When  units  simply  are  named,  units  of  the  first  order  are 
always  meoTit. 


15.  Have  we  a  separate  character  for  ten  1  How  do  we  express  ten  ' 
To  how  many  units  1  is  1  ten  equal  1  May  ten  be  regarded  as  a  single 
unit  !     Of  what  order  1 

IG.  When  two  figures  are  written  by  the  side  of  each  other,  what  place 
does  the  right  hand  figure  occupy  1  The  figure  on  the  left  1  ^\' hen 
units  simply  are  named,  what  unita  are  meant  ! 


Li  NOTATION. 

17.  In  order  to  express  ten  units  of  the  second  order,  or  one 
hundred,  we  form  a  new  combination : 

It  is  done  thus,         -----         100, 

by  writing  two  ciphers  on  the  right  of  1.  This  number  is  read, 
cue  hundred. 

li^ow,  this  one  hundred  expresses  10  U7iits  of  the  second  order, 
or  100  U7iits  of  the  first  order.  The  one  hundred  is  but  an 
individual  hundred,  and,  in  this  light,  may  be  regarded  as  a 
unit  of  the  third  order. 

We  can  now  express  any  number  less  than  one  thousand. 

For  example,  in  the  number  two  hundred  and  fifty- 
five,  there  are  5  unit?;.  5  tens,  and  2  hundreds.  Write, 
therefore.  5  units  of  the  first  order,  5  units  of  the  second     §    c   'S 

order,  and  2  of  the  third  ;  and  read  from  the  risht,  units.     „    "^     ^ 
'  >  -''255 

tens^  hundreds. 

In  the  number  five  hundred  and  ninety-five,  there  are  "  «  2 

5  units  of  the  first  order,  9  of  the  second,  and  five  of  the  ^  5  a 

third  ,•  and  it  is  read  from  the  right,  units,  tens,  hundreds.  5  9  5 

In  the  number  six  hundred  and  four,  there  are  4  units  '^  x  Z 

of  the  first  order,  0  of  the  second^  and  6  of  the  third.  ^  5  § 

TJie  right  hand  figure  ahvays  expresses  units  of  the 
first  order  ;  the  second,  units  of  the  second  order  ;  and  the  third, 
units  of  the  third  order. 

18.  To  express  ten  units  of  the  third  order,  or  one  thousand, 
we  form  a  new  combination  by  writing  three  ciphers  on  the 
right  of  1  ;  thus, 1000. 

Now,  tliis  is  but  one  single  thousand,  and  may  be  regarded  as 
a  unit  of  the  fourth  order. 

17.  How  do  you  write  one  hundred  ?  To  how  many  units  of  the  second 
order  is  it  equal  ?  To  liow  many  of  the  first  order !  How  may  it  be 
roiiardcd  1  Of  what  order  1  How  many  units  of  the  third  order  in  200  ! 
In  600  !      fii  900  ! 

18  To  what  are  ten  units  of  the  tiiird  order  equal  1  Hew  do  you  write 
it  1  How  do  you  write  a  single  unit  of  the  first  order  7  How  do  you  writo 
a  unit  of  the  second  order  1     Of  the  third  T     Of  the  fourth  \ 


NOTATION.  19 

Thus,   we   may  form  as  many  orders  of  units  as  we  please : 

a  single  unit  of  ihe  first  order  is  expressed  hy  -  -  1, 

a  unit  of  the  second  order  by  1  and  0;  tlius,  -  -  10, 

a  unit  of  tlie  third  order  by  1  and  two  O's ;  -  -  100, 

a  unit  of  the  fourth  order  by  1  and  three  O's  ;  -  -  1000, 

a  unit  of  the  fifth  order  by  1  and  four  O's  ;  -  -  10000  ; 
and  so  on,  for  units  of  higher  orders  : 

19.  Therefore, 

1st.  The  same  jrgure  expresses  different  units  according  to  the 
place  tc/n'ch  it  occupies  : 

2d.  Units  ofthejirst  order  occupy  the  place  at  the  right  ;  units 
of  the  second  order,  the  second  place  ;  units  of  the  third  order,  the 
third  place,  and  so  on  for  places  still  to  the  left: 

3d.  Ten  units  of  the  first  order  make  one  of  the  second  ;  ten  of 
the  second,  one  of  the  third ;  ten  of  the  third,  one  of  the  fourth  ; 
and  so  on  for  the  higher  orders  : 

4tli.  IVhen  figures  are  written  by  the  side  of  each  other,  ten 
units  in  any  one  place  male  one  unit  of  the  place  next  at  tlie 
left. 

EXAHPLKS    IN    WRITING    THE    ORDERS    OP   UNITS. 

1.  Write  7  units  of  the  1st  order. 

2.  Write  8  units  of  tlie  2d  order. 

3.  Write  9  units  of  the  4th  order. 

4.  Write  3  units  of  the  1st  order,  with  9  of  the  2d. 

5.  Write  9  units  of  the  3d  order,  with  6  of  the  2d,  and  1  of 
the  1st. 

G.  Write  0  units  of  the  2d  order,  8  of  the  1st,  with  4  of  the 
3d,  and  7  of  the  4th. 


19.  On  what  does  the  unit  of  a  figure  depend  1  What  is  the  unit  of 
the  place  on  the  right  1  What  is  the  unit  of  the  second  place  1  What  of 
the  third  place  1     What  of  the  fourth  1   &c. 

How  many  units  of  the  first  order  make  one  of  the  second  1  IIo';» 
many  of  the  second  make  one  of  the  third  ]  How  many  of  the  third  one 
of  the  fourth  1  &c.  When  figures  are  written  by  the  side  of  each  oilier, 
how  many  units  of  any  place  make  one  unit  of  the  place  next  at  the 
\eft  1 


20  NOTATION. 

7.  "Write  8  units  of  the  6tli  order,  7  of  the  4th,  9  of  the  5th, 
0  of  the  od,  2  of  the  2d,  and  1  of  the  1st. 

8.  'Write  8  units  of  the  8th  order,  6  of  the  7th,  0  of  the  1st, 
3  of  tlie  2a,  4  of  the  3d,  9  of  the  4th,  0  of  the  Gtli,  and  2  of 
the  5th. 

9.  AVrite  4  units  of  the  10th  order,  8  of  the  7th,  3  of  the  9th, 
2  of  tlie  8th,  0  of  the  6th,  3  of  the  1st,  G  of  the  2d,  0  of  the 
Sd,  1  of  the  4th,  and  2  of  the  5th. 

10.  Write  3  units  of  the  2d  order,  2  of  the  1st,  9  of  the  3d, 
0  of  the  4th,  9  of  the  9th,  G  of  the  8th,  7  of  the  7th,  0  of  the 
Gth,  and  4  of  the  5th. 

11.  Write  3  units  of  the  11th  order,  0  of  the  10th,  8  of  the 
4th,  0  of  the  5th,  2  of  the  Gth,  0  of  the  7th,  3  of  the  8th,  4  of 
the  9tli,  1  of  tlie  3d,  %  of  the  2d,  and  3  of  the  1st. 

12.  Write  3  units  of  the  12th  order,  G  of  the  11th,  3  of  the 
8th,  7  of  the  Gth,  2  of  the  4th,  and  1  of  the  2d. 

13.  Write  5  units  of  the  13th  order,  8  of  the  12th,  0  of  the 
9th,  G  of  tlie  7th,  8  of  the  3d,  and  12  of  the  1st. 

14.  Write  7  units  of  the  14th  order,  5  of  the  13th,  G  of 
the  12th,  5  of  the  10th,  7  of  the  8th,  9  of  the  Gth,  5  of  the 
4th,  and  8  of  the  1st. 

15.  Write  9  units  of  the  15th  order,  4  of  the  13th,  8  of 
the  9 til,  2  of  the  Gth,  7  of  the  3d,  and  2  of  the  2d. 

IG.  Write  G  units  of  the  IGth  order,  9  of  the  12th,  7  of  the 
9th,  4  of  the  7th,  0  of  the  Gth,  8  of  the  4th,  9  of  the  5th,  and 
2  of  the  2d. 

17.  Write  8  units  of  the  20th  order,  5  of  the  18th,  G  of 
the  13th,  4  of  the  11th,  9  of  the  9th,  1  of  the  17th,  4  of  the 
5th,  and  9  of  the  3d. 

18.  Write  6  units  of  the  10th  order,  5  of  the  8th,  9  of  the 
Gth,  0  of  the  4th,  and  1  of  the  1st. 

19.  Write  9  units  of  the  18th  order,  and  then  diminish  the 
figure  of  each  order  by  1  till  you  come  to  and  include  0  ;  then 
increase  the  figure  of  each  order  by  1,  till  you  reach  the  first 
order; and  then  read  each  order. 

20.  Write   tJie   number  which  has  20  units  of  the  17tli  order, 


NUMEEATION. 


21 


0  ol  the  14tli,  8  of  the  16th,  4  of  the   13th,  0  of  the  8th,  0  of 
the  9th,  and  one  in  each  of  the  other  places. 

NUMERATION. 

20.  Numeration  is  the  art  of  reading  correctly  any  number 
expressed  by  figures  or  letters. 

The  pupil  has  already  been  taught  to  read  all  numbers  from 
one  to  one  thousand.  The  Numeration  Table  will  te^ch  him  to 
read  any  number  whatever;  that  is,  to  express  numbers  in 
words. 

TABLE. 

7th  Period.     6th  Period.     5th  Period.     4th  Period.  3d  Period.     2d  Period.      1st  Period. 
Quintiilioiis.  Quadrillions.  Trillions.        Billions.     Millions.     Thousands.        Units. 


09 

s    -    • 

o 

. 

, 

« 

W 

• 

o 

^<-j 

m 

X/2 

^      DQ        ' 

."  <» 

s  ■ 

W 

r* 

• 

rt      .      • 

I        • 

t-     !=< 

_o 

f-; 

o 

W      OT 

s-i  • 

n3    O      . 

O 

•i-H            ^ 

w 

o  a 

• 

o-S   • 

G"!  : 

H-2 

..— (     (—1 

P3   3 

^ 

o 

E— 1     <« 

"—  -iS    -A 

tt-  rt  » 

c*.^   , 1 

V-.  '.^ 

t— 

"~^ 

u^    o 

0    =    5 

^  §  s 

O'm 

o  'z3 

o 

•  — 1 

=>-    m 

• 

K  C?"  3 

^a° 

«=H 

Kpq 

w 

cc  H  'Cl 

M 

^     O     i^ 

go 

_o 

1^ 

o 

Cm 
O 

a; 

a 
o 

1^  1 

4) 

• 

S      J^      TO 

s  J2 

•3 

T3    cc 

O 

r^ 

CO 

a  s  § 

c  J2  -2 

D    o  '3 

C3   a>   ^ 

s  § 

S-l 

3     O 

f^ 

^ 

o 

s 

a    O  -3 

p  o  '3 

KHC^ 

KHGf 

ffiHH 

KHpq 

H 

KHH 

EH^ 

3   7   0 

8  9   4 

2   1 

6 

6  3 

6 

8 

0 

6 

3   0  4 

6  2 

5 

Notes. — 1.  Numbers  expressed  by  more  than  three  figures  are 
written  and  read  by  periods^  as  shown  in  the  above  table. 

2.  Each  period  always  contains  three  figures,  except  the  left  hand 
period,  which  may  contain  one,  two  or  three  figures. 

3.  The  unit  of  the  first,  or  right-hand  period,  is  1  ;  of  the  second 
period.  1  thousand  ;  of  the  third,  1  million ;  of  the  fourth,  1  billion  ; 
and  so  on,  for  periods,  still  to  the  left. 

4.  To  Quintillions  succeed  Sextillions,  Septillions,  Octillions,  Non- 
lllions,  Decillions,  UiidecilIion.=5,  Daodecillions,  &c. 

5.  The  pupil  should  be  required  to  commit,  thoroughly,  the  names 

20.  What  is  Numeration  1  "What  is  the  unit  of  the  first  period  1  Wha} 
is  the  unit  of  the  second  1  Of  the  third  1  Of  the  fourth  1  Fifth  ?  Sixth  T 
Seventh  1  &c.  Give  the  rule  for  reading  numbers. '  Give  the  rule  for 
writing  numbers. 


22 


NUMEKATION. 


of  the  periods,  so  as  to  repeat  tlicm  in  their  regular  order  from  left 
to  right,  as  well  as  from  right  to  lefc. 

6.  Formerly,  in  the  English  Notation,  six  places  Avere  given  to 
Millions,  Trillions,  Quadrillions.  &:c.  They  were  read,  Millions,  Tens 
of  Millions,  Hundreds  of  Millions,  Thousands  of  Millions,  Tens  of 
Thousands  of  Millions,  Hundreds  of  Thousands  of  Millions  ;  and  the 
same  for  Billions,  Trillions,  Quadrillions.  Sec.  This  method  produc- 
ed great  irregularity  in  the  Notation,  as  it  gave  three  places  to  the 
units  of  the  first  two  periods,  (viz. :  units  and  thousands,)  and  six 
places  to  each  of  the  others.  The  French  method,  which  gives  three 
places  to  the  unit  of  each  period,  is  fully  adopted  in  this  country,  and 
must  soon  become  universal. 

RULE    FOR    READING   NUMBERS. 

I.  Separate  the  number  into  periods  of  three  ficjures  each, 
beginning  at  the  riijht'hand. 

II.  Name  the  unit  of  each  figure,  beginning  at  the  right  hand. 

III.  T7ie7i,  beginning  at  the  left  hand,  read  each  period  as  if 
it  itood  alone,  naming  its  unit. 


EXAMPLES    IN    READING    NUMBERS. 

Let  the  pupil  point  oflf  and  read  the  following  numbers — then 
write  them  in  words. 


1. 

97 

6. 

32045607 

11.      784236704 

2. 

326 

7. 

90464213 

12.     7403026054 

3. 

3302 

8. 

47364291 

13.    217040S0495 

4. 

65042 

9. 

4037902169 

14.    21896720421 

5. 

742604 

10. 

91046302 

15.  8140290308097 

16. 

8504680467023 

19.    30467214302704 

17. 

90403040720156 

20.   167320410341204 

18. 

1723047368 

93210 

2 

1.  2164032189765421 

Let  each  of  the  above  examples,  after  being  written  on  tliQ 
\)lack  board,  be  analyzed  as  a  class  exercise ;  thus  : 

1.  In  how  many  ways  may  the  number  97  be  read? 
1st.  The  common  way,  97. 
2d.  Wo  may  read,  9  tens,  and  7  units. 


NATUKE    OF    NUMBEKs,  23 

2.  Ill  how  many  ways  may  32 G  be  read  ? 

1st.   By  the  common  way,  tliree  hundred  and  twenty-six. 
2d.  Three  hundreds,  2  tens,  and  6  units. 
3d.  Tliirty-two  tens,  and  six  units. 

3.  In  how  many  ways  may  the  number  5302  be  read? 
1st.  Five  tliousand  three  hundred  and  two. 

2d.  Five  tliousand,  tliree  Iiundreds,  0  tens,  and  2  units. 
3d.  Fifty-three  hundreds,  0  tens,  and  2  units. 
4th.  Five  hundred  and  tliirty  tens,  and  2  unita. 

4.  In  65042,  liow  many  ten  thousands  ?  How  many  thou- 
sands ?  How  many  hundreds  ?  IIow  many  tens  ?  How  many 
units  ? 

5.  In  742604,  how  many  hundred  thousands?  How  many 
ten  tliousands  ?  How  many  thousands  ?  How  many  hundreds  ? 
How  many  tens  ?     IIow  many  units  ? 

RULE    FOR    WRITING    NUMBERS,    OR    NOTATION. 

I.  Begin  at  the  left  hand  and  write  each  period  in  order,  as  if 
it  were  a  period  of  units. 

II.  When  the  nuynher,  in  any  period  except  the  left-hand  period, 
can  he  expressed  hy  less  than  three  fgures,  prefix  one  or  two 
ciphers  ;  and  when  a  vacant  period  occurs,  fill  it  with  ciphers. 

EXAMPLES    IN   NOTATION. 

Express  the  following  numbers  in  figures  : 

1.  Six  hundred  and  twenty-one. 

2.  Five  thousand  seven  hundred  and  two. 

3.  Plight  thousand  and  one. 

4.  Ten  thousand  four  hundred  and  six. 

5.  Sixty-five  thousand  and  twenty-nine. 

6.  Forty  millions  two  hundred  and  forty-one. 

7.  Fifty-nine  millions  tliree  hundred  and  ten. 

8.  Fleven  thousand  eleven  hundred  and  eleven. 

9.  Three  hundred  millions,  one  thousand  and  six. 

10.  Sixty-nine    billions,    three   millions,    two    hundred    and 
eleven. 


24  DEKOMINATh    NUMBERS. 

11.  Forty-seven  quadrillions,  sixty-nine  billions,  four  bun- 
dred  and  sixty-five  thousands,  two  hundred  and  seven. 

12.  Eight  hundred  quintillions,  four  hundred  and  twenty-nine 
millions,  six  thousand  and  nine. 

13.  Ninety-five  sextiliions,  eighty-nine  millions,  eighty-nine 
thousands,  three  hundred  and  six. 

14.  Six  quintillions,  four  hundred  and  fifty-one  billions,  sixty- 
five  millions,  forty -seven  thousands,  and  one  hundred  and  four. 

15.  Write,  in  figures,  nine  hundred  and  ninety-nine  billions, 
sixty-five  millions,  eight  hundred  and  forty-one  thousands,  four 
hundred  and  eleven, 

1 G.  Four  hundred  and  seventy  nonillions,  forty  octillions,  four 
millions,  six  thousands,  two  hundred  and  four. 

17.  Sixty-five  sextiliions,  eight  hundred  quadrillions,  seven 
hundred  and  fifty  billions,  seven  hundred  and  fifty-one  millions, 
nine  hundred  and  seventy-five  thousands,  three  hundred  and 
ten. 

FORMATION  AND  NATURE  OF  NUMBERS. 

21.  The  term,  one,  may  refer  to  any  single  thing :  it  has  no 
reference  to  kind  or  quality :  it  is  called  an  abstract  unit. 

22.  The  term,  one  foot,  refers  to  a  single  foot,  and  is  called 
a  concrete  or  denominate  unit. 

23.  An  abstract  number  is  one  whose  unit  is  abstract :  thus, 
three,  four,  six,  &c.,  are  abstract  numbers. 

24.  A  concrete  or  denominate  number,  is  one  whose  unit  is 
concrete  or  denominate :  thus,  three  feet,  four  dollars,  five 
pounds,  are  denominate  numbers. 

25.  A  Simple  Number  is  a  single  unit,  or  a  single  collection 
of  units,  either  abstract  or  denominate. 


21.  Docs  the  term,  one,  refer  to  the  kind  of  thing  to  which  it  is  applied  1 
What  is  it  called? 

22.  To  what  docs  one  foot  refer  1     What  is  it  called  ' 

23.  What  is  an  abstract  number? 

24.  What  is  a  concrete,  or  denominate  number  1 

25.  What  is  a  simple  number' 


DENOMINATE   NUMBERS.  25 

26.  Quantity  is  anything  which  can  be  measured. 

27.  Numbers  which  have  the  same  unit  are  of  the  same 
denomination  :  and  numbers  having  different  units  are  of  differ- 
ent denominations.  Tims,  4  yards  and  6  yards  are  of  the  same 
denomination  i  but  4  yards  and  G  feet  are  of  different  denomi- 
nations. 

28.  If  two  or  more  denominate  numbers,  having  different 
units,  are  connected  together,  forming  a  single  expression,  this 
is  called,  a  compound  denominate  numbei*.  Thus,  3  yards  2 
feet  and  6  inches,  is  a  cotnpotind  denominate  number. 

29.  We  have  seen  (Art.  19)  that  when  figures  are  written 
by  the  side  of  each  other,  thus, 

678904, 
the  language  implies  that  ten  units  of  any  place  make  one  unit 
of  the  place  next  to  the  left. 

30.  When  figures  are  written  to  express  English  Currency, 
thus:  £         s.         d.        far. 

4        17        10         3 
the  language  implies,  that  four  units  of  the  lowest  denomination 
make  one  of  the  second  ;  twelve  of  the  second,  one  of  the  third  ; 
and  twenty  of  the  third,  one  of  the  fourth. 

31.  "When  figures  are  written  to  express  Avoirdupois  weight, 
thus :  T.     cwt.     qr.     lb.     oz.     dr. 

27      17       2      24     11     10 
the  language  implies,  that  16  units  of  the  lowest  denomination 

26.  What  is  quantity  \ 

27.  When  are  numbers  paid  to  be  of  the  same  denomination  1  When 
of  different  denominations  ! 

28.  What  is  a  compound  denominate  number  1 

29.  When  several  figures  are  simply  written  by  the  side  of  each  other, 
what  docs  the  language  imply  1 

33.  In  the  English  Currency,  how  many  units  of  the  lowest  denomina- 
tion make  one  of  the  second  '\  How  many  of  the  second  one  of  the  third  ! 
How  many  of  the  third  one  of  the  fourth ! 

31.   In  the  Avoirdupois  weight,  how  many  of  the  lowest  make  ^ 
the  second  1     How  many  of  the  second  one  of  the  third  ? 

2 


26  DENOMINATE   NUMBEKS. 

make  one  of  the  second;  16  of  the  second,  one  of  the  third; 
25  of  the  third,  one  of  the  fourth ;  4  of  the  fourth,  one  of  the 
6fth ;  and  20  of  the  fifth,  one  of  the  sixth. 

All  the  other  compound  denominate  numbers  are  foi'med  on 
the  same  principle  ;  and  in  all  of  them,  we  pass  from  a  lower 
to  the  next  higher  denomination  hy  considering  how  many  units 
of  the  lower  make  one  unit  of  the  next  Jiigher* 

32.  A  Scale  expresses  the  relations  between  the  different 
units  of  a  number.  There  are  two  kinds  of  scales — uniform 
and  varying.  In  the  common  scale,  the  number  of  units  which 
make  1  of  the  next  higher  is  10.  In  English  Currency,  4,  12, 
and  20,  make  up  the  varying  scale ;  and  16,  16,  25,  4  and  20, 
in  Avoirdupois  weight. 

SCALE   OF   TENS. 

33.  Let  us  write  a  row  of  O's,  thus  ; 


o 


rag         S  .2         H 


a 


o 


rt      p 


J^or:^        '-'^ud        So,^        J^ort 
GHPq      KHS      KHH      SHP 

0  0  0,      0   0  0,      0  0   0,     0    0  0, 

The  language  of  figures  determines  the  unit  of  each  place,  and 
also,  the  laip  of  change  in  passing  from  one  place  to  another. 
This  is  called  the  decimal  system,  in  v»'hich  the  units  change 
according  to  the  scale  of  tens. 

If  it  be  required  to  express  a  given  number  of  units,  of  any 

*  For  the  Tables  of  Denominate  Numbers,  see  Appendix,  page  383. 

32.  What  is  a  scale  1  How  many  kinds  of  scales  are  there'?  "What 
are  they  ?  What  is  the  scale  in  the  common  system  of  numbers  !  ^^'llat 
is  tlie   scale  in  English   Currency  1      What  in  Avoirdupois  weight  ? 

33.  If  a  row  of  O's  be  written,  what  does  the  language  of  figures  deter- 
mine ?  What  is  such  a  system  called  1  How  does  the  unit  change' 
How  do  you  express  a  given  number  of  units  of  any  order  ? 


INTEGRAL    UNITS.  27 

Older,  we  first  select  from  tlie  arithmetical  alphabet  the  figure 

which  designates  tlie  number,  and  then  write  it  in  the  place 

corresponding  to  the  order.     Thus,  to  express  three  millions, 

we  write 

3000000; 

and  similarly  for  all  numbers. 

UNITED  STATES  MONEY. 
34.  United  States  money  atlbrds  an  example  of  a  system  of 
denominate  units,  increasing  according  to  the  scale   of  tens: 
thus, 

^       a       (O      ^    _r- 
Of)    ;z5       g      S     " 

W     Q     P     O     S 

11111 

may  be  read  11  thousand  1  hundred  and  11  tnills ;  or,  1111 
cents  and  1  mill;  or,  111  dimes,  1  cent,  and  1  mill;  or,  11  dol- 
lars, 1  dime,  1  cent,  and  1  mill ;  or,  1  eagle,  1  dollar,  1  dime, 
1  cent,  and  1  mill.  Thus,  we  may  read  the  number  with  any 
one  of  its  units  as  a  base,  or  we  may  name  them  all ;  as 
1  eagle,  1  dollar,  1  dime,  1  cent,  1  mill.  Generally,  in  United 
States  money,  we  read  in  the  denominations  of  dollars  cents 
and  mills  ;  and  say,  11  dollars  11  cents  and  1  mill. 

United  States  money  is  denoted  by  the  character,  $.  The 
figures  expressing  dollars  are  separated  from  those  which  denote 
cents  and  mills  by  a  comma ;  thus, 

$11,111 
is  read,  11  dollars  11  cents  1  mill;  the  figures  on  the  left  of 
the   comma  always  denote   dollars  ;  the  first  two  on  the  right 
denote  cents,  and  the  third,  mills. 

ALIQUOT   TARTS. 

One  number  is  said  to  be  an  aliquot  part  of  another,  when 

84.   Are  the  numbers   used  in   United  States  nionej'  abstract  or  denomi- 
nate ?     According  to   what  scale  do  the  units  change  1      How  are  <^ 
separated  from  cents  and  mills  I     A\  hat  is  an  rdiquot  parti 
aliquot  parts  of  a  dollar  ' 


28 


VARYING   SCALKS. 


it  is  contained  in  that  other  an  exact  number  of  times.  Thus ; 
50  cents,  25  cents,  &c.,  are  aliquot  parts  of  a  dollar :  so  also, 
2  months,  3  months,  4  months  an.l  6  months  are  aliquot  parts 
of  a  year.  The  parts  of  a  dollar  are  sometimes  expressed  frac- 
tionally, as  in  the  following 

TABLE, 
cents. 


$1 

i  of  a  dollar  = 


100 
50    cents. 
1  of  a  dollar  =     33 J  cents. 

cents. 
20    cents. 


1-  of  a  dollar  =     25 
1  of  a  dollar  = 


l  of  a  dollar 
Jq  of  a  dollar 


121  cents. 
10    cents. 
J^  of  a  dollar  =     61  cents. 
5    cents. 
5    mills. 


o\j^  of  a  dollar 


J    of  a  cent 


VAKYING    SCALES. 

35.  If  we  write  the  well-known  signs  of  the  English  currency, 
and  place  1  under  each  denomination,  we  shall  have 

£     s.    d.  far. 
1111 

Now,  the  signs  £.s.  d.  and  far.  fix  the  value  of  the  unit  1  in 
each  denomination  ;  and  they  also  determine  the  relations 
between  the  different  units.  For  example,  this  simple  language 
expresses  the  following  ideas  : 

1st.  That  the  unit  of  the  right  hand  place  is  1  farthing — of 
the  place  next  at  the  left,  1  penny — of  the  next  place,  1  shilling 
— of  the  next  place,  1  pound  ;  and 

2c?.  That  4  units  of  the  lowest  denomination  make  one  unit 
of  the  next  higher;  12  of  the  second,  one  of  the  third;  and 
20  of  the  third,  one  of  the  fourth.  Hence,  4,  12  and  20,  make 
up  the  scale. 

36.  If  we  take  the  denominate  numbers  of  Avoirdupois 
weight,  we  have 

Ton    act.     qr.      lb.     oz.      dr. 
111111; 

35  In  En^lisli  currency,  is  the  scale  uiiiforni  or  varying  ?  How  does 
it  vary  1 

30.  Name  the  units  of  the  scnle  in  AvoirJupois  weii^Iit. 


INTEGRAL   UNITS.  29 

in  which  the  units  increase  in  the  following  manner ;  viz. : 
counting  from  the  right,  IG  units  of  the  lowest  denomination 
make  1  unit  of  the  second;  IG  of  the  second,  1  of  the  third; 
25  of  tlie  third,  1  of  tlie  fourth  ;  4  of  the  fourth,  1  of  the  fifth  ; 
20  of  the  fifth,  1  of  the  sixth.  The  scale,  therefore,  for  this 
class  of  denominate  numbers,  varies  according  to  the  above 
law. 

37.  If  we  take  any  other  class  of  denominate  numbers,  as 
Troy  weight,  or  any  of  the  systems  of  measures,  Ave  shall 
have  different  scales  for  the  formation  of  the  dilFerent  numbers. 
But  in  all  the  formations,  we  shall  recognize  the  application  of 
the  same  general  principles. 

There  are,  therefore,  two  general  methods  of  forming  the 
different  systems  of  integral  numbers  from  the  unit  one.  The 
first  consists  in  preserving  a  uniform  law  of  relation  between 
the  different  units.  If  that  law  of  relation  is  expressed  by  10, 
we  have  the  system  of  common  numbers. 

The  second  method  consists  in  the  application  of  known, 
though  varying  laws  of  change  in  the  units.  These  changes  in 
the  units,  produce  difl'erent  systems  of  denominate  numbers, 
each  of  which  has  its  appropriate  scale. 

INTEGRAL  UNITS  OF  ARITHMETIC. 

38.  The  Integral  units  of  Arithmetic  may  be  divided  into 
ei2;ht  classes  : 

1st.  Abstract  units  :  2d.  Units  of  currency  :  3d.  Units  of 
length  :  4:ih.  Units  of  surface  :  5th.  Cubic  units,  or  units  of 
volume  :  (J/h.  Units  of  weight :  7ih.  Units  of  time :  8lh.  Units 
of  circular  measure. 

First  among  the  units  of  arithmetic  stands  the  abstract  unit  1. 
This  is  the  primary  base  of  all  abstract  numbers,  and  becomes 
the  base,  also,  of  all  denominate  numbers,  by  merely  naming, 
in  succession,  the  particular  thing  to  which  it  is  applied. 

37.  How  many  general  methods  are  there  of  forming  numbers  from  tlie 
unit  one  ]     What  is  the  first  ?     WTiat  is  the  second  1 

38.  Into  liow  many  general  classes  may  the  units  of  Arithmetic  bo 
arranged  ^     What  are  they  1 


30  rr.oPERTiES  of  9's. 

OF    THE    SIGNS. 

39.  The  sign  ~,  is  called  the  sign  o^  equality.  When  placed 
between  two  numbers  it  denotes  that  they  are  equal ;  that  is, 
that  each  contains  the  same  number  of  units. 

The  sign  +,  is  called  p^us,  which  signifies  jnore.  When  placed 
between  two  numbers  it  denotes  that  they  are  to  be  added 
together :  thus,  3  +  2  =  5. 

The  sign  — ,  is  called  minus,  a  term  signifying  less.  When 
placed  between  two  numbers  it  denotes  that  the  one  on  the 
right  is  to  be  taken  from  the  one  on  the  left :  thus,  6  —  2  =  4. 

The  sign  x ,  is  called  the  sign  of  multiplication.  When 
placed  between  two  numbers  it  denotes  that  they  are  to  be  mul- 
tiplied together;  thus,  12  X  3,  denotes  that  12  is  to  be  multi- 
plied by  3. 

The  parenthesis  is  used  to  indicate  that  the  sum  of  two  or 
more  numbers  is  to  be  considered  as  a  single  number  :  thus, 

(2  4-  3  +  5)  X  G 
shows,  that  the  sum  of  2,  3  and  5  is  to  be  multiplied  by  6. 

The  parenthesis  is  also  used  to  denote  that  the  difference  be- 
tween two  numbers  is  to  be  considered  as  a  single  number ;  thus, 

(5  -  3)  X  6, 
denotes  that  the  difference  between  5  and  3  is  to  be  multiplied 
by  6. 

The  sign  -r-,  is  called  the  sign  of  division.  When  placed 
between  two  numbers  it  denotes  that  the  one  on  the  left  is  to  be 
divided  by  the  one  on  the  right :  •  thus,  4-^-5,  denotes  that  4  is 
to  be  divided  by  5. 

PKOPERTIES    OF    THE    9's. 

40.  In  any  number,  written  with  a  single  significant  figure, 
as  4,  40,  400,  4000,  &;c.,  the  excess  over  exact  9's  is  equal  to 

39.  What  is  the  sign  of  equaHty  \  AVliat  is  the  sign  of  adJilioii  ? 
Wiiat  of  siil)traction  1  What  of  niuhipHcation  !  For  wliat  is  tlie  i)aien- 
thesis  used  !     What  is  Ihe  sign  of  division  1 

40.  What  will  be  the  excess  over  exact  9's  in  any  number  expressed  by 
a  single  significant  figure  1  How  may  the  excess  over  exact  9's  be  found 
in  any  number  whatever'! 


REDUCTION. 


31 


the  number  of  units  in  the  significant  figure.     For,  any  such 

number  may  be  written, 

4  =  4. 

Also,  -        -        -        -  40  =  (9       -f  1)  X  4, 

«      -        -        -        -  400  =  (99    +  1)  X  4, 

«      -        -         -        -         4000  r=  (999  +  1)  X  4, 
&c.,  &c.,  &c. 

Each  of  the  numbers  9,  99,  999,  &c.,  contains  an  exact  number 
of  9's  ;  hence,  when  muUipliecl  by  4,  the  several  products  will 
contain  an  exact  number  of  9's  :  therefore, 

TJie  excess  over  exact  9's,  in  each  mmiber,  is  4  ;  and  the  same 
may  be  shotvn  for  each  of  the  other  significant  figures. 
If  we  write  any  number,  as 

6253, 
we  may  read  it  6  thousands  2  hundreds  5  tens  and  3.  Now, 
the  excess  of  9's  in  the  0  thousands  is  6  ;  in  2  hundreds  it  is  2  ; 
in  5  tens  it  is  5  ;  and  in  3  it  is  3  :  hence,  in  them  all,  it  is  16, 
which  is  one  9  and  7  over :  therefore,  7  is  the  excess  over 
exact  9's  in  the  number  6253.     In  like  manner, 

The  excess  over  exact  9's  in  any  number  whatever,  is  found  by 
addi?ig  together  the  sign'ficmit  fgurcs  and  rejecting  the  exact  9's 
from  the  sum. 

Note. — It  is  best  to  reject  or  drop  the  9  as  soon  as  it  occurs  :  thus 
we  say,  3  and  5  are  8  and  2  are  10  ;  then,  dropping  the  9,  we  f^ay, 
1  to  6  is  7,  which  is  the  excess  ;  and  the  same  for  all  similar  operations. 

1.  What  is  the  excess  of  9's  in  48701  ?     In  €7498  ? 

2.  What  is  the  excess  of  9"s  in  9472021  ?     In  2704962  ? 

3.  What  is  the  excess  of  9's  in  87049612  ?     In  4987051  ? 

REDUCTION. 

CHANGE    OF     UNITS. 

41.  Reductiox  is  the  operation  of  changing  the  unit  of  a 
number  without  altering  its  value.     Thus,  if  we  have  4  yards, 


41.  What  is  Reduction  1  How  do  you  change  yards  to  feet  1  How  do 
you  change  feet  to  inches  1  How  do  you  change  inches  to  feet  1  How  do 
you  change  feet  to  yards  '' 


/ 


32  RKDCCTION. 

in  whicli  the  unit  is  1  yard,  and  wisli  to  change  to  feer,  the 
units  of  the  scale  will  be  3,  since  3  feet  make  1  yard :  there- 
fore, the  number  of  feet  will  be 

4x3  =  12  feet. 
If  it  were  required  to  reduce  12  feet  to  inches,  the  units  of  the 
scale  would  be  12,  since  12  inches  malie  1  foot.     Hence, 

4  yards  =z  4  x  3  =z  12  feet  =  12  x  12  =  144  inches. 

H',  on  the  contrary,  we  wash  to  change  144  inches  to  feet,  and 
then  to  yards,  we  should  first  divide  by  12,  the  units  of  the 
scale  in  passing  from  inches  to  feet ;  and  then  by  3,  the  units 
of  the  scale  in  passing  from  feet  to  yards.  Hence,  Reduction 
is  of  two  kinds  : 

1st.  To  reduce  a  number  from  a  higher  unit  to  a  lower: 
Multiply  tlie  units  of  the  Idgliesl  denomination  by  the  number 
of  units  ill  the  scale  which  connect-^  it  with  the  next  lower,  and 
then,  add  to  t  e  product  the  un'ts  <f  that  deno  mi  nation :  Pro- 
ceed in  the  same  manner  through  all  the  denominations  till  the 
unit  is  brought  to  the  requird  denomm'ttion. 

Id.  To  reduce  a  number  from  a  lower  unit  to  a  higher: 
Divide  the  given  number  hy  the  number  of  units  in  the  scale 
which  connects  it  with  the  next  higher  denomination  ;  and  set 
down  the  remainder,  if  there  be  one.  Divide  the  quotient  thus 
obtained,  and  each  succeeding  quotient  in  the  satne  manner,  till 
the  unit  is  reduced  to  the  required  denomination:  the  last  quo- 
tient with  the  several  remainders  annexed,  will  be  the  answer. 

EXAMPLES. 

1.  Reduce  £3  14s.  4c?.  to  pence.  "We  first  multiply  the  £3 
by  20,  which  gives  GO  shillings.  We  then  add  14,  making  74 
shillings  :  we  next  multiply  by  12,  and  the  product  is  888  pence  : 
to  this  we  add  4rf.  and  we  have  892  pence,  which  are  of  the 
same  value  as  £3  14s.  Ad. 

If,  on  the  contrary,  we  wished  to  change  892  pence  to  pounds 
Bhillings  and  pence,  we  should  first  divide  by  12:  the  quotient 
is  74  shillings,  and  4c?.  over.     We  next  divide  by  20,  and  the 


REDUCTION. 


33 


quofient  is  £3,  and  14s.  over:  hence,  the  rebuilt  is  £3  I'ls.  4c?., 
which  is  equal  to  892  pence. 

The  reductions,  in  all  the  denominate  numbers,  are  made  in 
the  same  manner. 


2.  In  £5  5s.,  how  many  shil- 
lings, pence,  and  farthings .'' 

£        s. 
5        5 

20 


105  5  shillings  added. 
12 


12G0 
4 

5040 


Here  the  reduction  is  from  a 
greater  to  a  less  unit. 

4.    In    342".,    IQcivt.,    oqrs., 
Idlb.,  how  many  pounds  ? 

34 

20 


3.  In  5040  farthings,  how  many 
pence,  shillings,  and  pounds  ? 

4)5040  farthings. 
12)1260  pence. 
2i0)IO|5~shillings. 
£5  OS. 


In  this  example,  the  reduc 
tion  is  from  a  less  to  a  greater 
unit. 


IGcwt.  added. 
oqr.  added. 
19  lb.  added. 


5.  In    GOQOAlb.,    how   many 
tons,  cwt.,  qr.,  and  lb. 

25)69G94 


A71S. 


AJ2787 qr.  . 
20)G06cwt.  . 
3342:    . 
34 r.  IGcwt. 


.  .  19lb. 
.  .  3qr. 
.  .  IGcwL 
Sqr.    19/5. 


696 

4 

2787 

25 

13954 

5574 

69694/^5. 

6.  In  $426,  how  many  cents  ?     How  many  mills  ? 

7.  In  36  eagles  8  dollars  and  6  dimes,  how  many  cents? 

8.  In  8750  mills,  hoAv  many  dollars  and  cents  ? 

9.  In  43  eagles  3  dollars  and  5  mills,  how  many  mills  ? 

10.  In  £37  9s.  8c?.,  how  many  pence? 

11.  In  1569    farthings,  how  many  pounds,  shillings,  pence, 
and  farthings  ?  2* 


34  REDUCTION. 

12.  In    IT.   Wciot.    Iqr.    2011/s.,   Avoirdupois,    how    many 
pounds  ? 

13.  In  15445^3.,  Avoidupois,  how  many  tons,  cwts.,  qrs.,  and 
lbs. 

14.  How    many  grains  of   silver  in  4Ib.,  Goz.,  12dwt.    and 
7grs.? 

15.  How  many  pounds,  ounces,  pennyweights,  and  grains  of 
gold,  in  704121  grains? 

16.  In  5lfe, 13,15,19,  2gr.,  Apothecaries'  weight,  how 
niany  grains? 

17.  In  174947  grains,  how  many  pounds,  ounces,  drachms, 
scruples  and  grains  ? 

18.  In  6  yards  2  feet  9  inches,  how  many  inches  ? 

19.  In  5  miles,  how  many  rods,  yards,  feet  and  inches  ? 

20.  In  2730  inches,  how  many  yards  feet  and  inches  ? 

21.  In  56  square  feet,  how  many  square  yards  ? 

22.  In  355  perches,  or  square  rods,  how  many  acres,  rooda 
and  perches  ? 

23.  In  456  square  chains,  how  many  acres  ? 

24.  In  BA.,  21i.,  8P.,  how  many  perches  ? 

25.  In  14  tons  of  round  timber,  how  many  cubic  mches  ? 

26.  In  31  cords  of- wood,  how  many  cubic  feet  ? 

27.  In  56320  cubic  feet,  how  many  cords  ? 

28.  In  157  yards  of  cloth,  how  many  nails  ? 

29.  In  192  Ells    Flem.,  how  many  yards? 

30.  07yd.,  Sqr.,  how  many  Ells  English  ? 

31.  In  A/thd.  Wine  measure,  how  many  quarts? 

32.  In  7560  pints,  Wine  measure,  how  many  liogsheads? 

33.  In  7  hogsheads  of  ale,  how  many  pints  ? 

34.  In  74304  half  pints  of  ale,  how  many  barrels  ? 

35.  In  31  bushels,  Dry  measure,  how  many  pints  ? 

36.  In  2110  pints,  Dry  measure,  how  many  bushels  ? 

37.  In   2   solar  years  of  365c?.  5h.  ASm.  4.8sec.,  each,  how 
many  seconds  ? 

38.  How  many  months,  weeks  and  days  in  254  days,  reckon- 
ing the  month  at  30  days  ? 


ADDITION.  35 


ADDITION. 

42.  The  sum  of  two  or  more  numbers  is  a  number  containing 
as  many  units  as  all  the  numbers  taken  together. 

Addition  is  the  operation  of  finding  the  sum  of  two  or  more 
numbers. 

1.  What  is  the  sum  of  769  and  487 

Analysis. — Write  the  numbers  thus  : 

di-aw  a  line  beneath  them,  -  -  .  - 
sum  of  the  units,  ----.. 
sum  of  the  tens,    - 

sum  of  the  hundreds,     .         .         -         -         - 
Entire  sura,  -         -         . 

The  example  may  be  done  in  another  way.  thus : 
Set  down  the  number  as  before  :  then  say,  7  and  9         operation. 
are  16  :  set  down  6  in  the  units  place,  and  the  1  ten  769 

under  the  8  in  the  column  of  tens.     Then  say,  1  to  8  487 

are  9,  and  6  are  15.     Set  down  the  5  in  the  column  n  6 

of  tens,  and  the  1  hundred  in  the  column  of  hundreds.  1256 

We  then  add  the  hundreds,  and  find  their  sum  to  be 
12  :  hence,  the  entire  sum  of  1256. 

Note  1 . — Observe,  that  units  of  the  same  value  are  always  written 
in  the  same  column,  since  every  collection  must  contain  units  of  the 
same  kind. 

2.  Wlien  the  sum  in  any  column,  exceeds  9,  it  produces  one  or 
more  units  of  a  higher  order,  which  belong  to  the  next  column  at  the 
left.  la  that  case,  write  down  the  excess  over  tens,  and  add  the  tens 
to  the  next  column.  This  is  called  carrying  to  the  next  column.  The 
number  to  be  carried,  should  not,  in  practice,  be  written  under  the 
column  at  the  left,  but  added  mentally. 

JBeginners,  however,  should    set    down  the  numbers  to  be  carried, 


42.  What  is  the  sum  of  two  or  more  numbers'!  What  is  Addition ! 
How  are  numbers  written  down  for  Addition  ?  What  do  you  do  in  simple 
numbers  when  the  sum  of  E.ny  column  exceeds  9  1  What  is  this  called  '* 
What  is  the  general  rule  for  the  addition  of  numbers  '^ 


3G  ADDITION. 

each  under  its  proper  columiij  as  in  the  examples  below. 

(2)  (3)  (4) 

85468  672143  4783614 

9104  79161  504126 

379  8721  872804 

94951         760025  6160544 

1012  12110  211101 

5.  What  is  the  sum  of  35  dollars  4  dimes  6  cents  5  mills, 

4  dollars  7  mills,  and  97  cents  3  mills  ? 

Note. — Write   the    units   of   the    same   value   in  o?eratio:m. 

the   same    column,  separating  the  dollars  from  the  $35,465 
cents  and  mills  by  a  comma  (Art.  40) :    then  add  the  4,007 

columns  as  m  simple  numbers.  ,973 

6.  Let   it   be   required   to   find    the    sum   ot  'm 
£14  7s.  Sd.  3far.,  and  £6  18s.  dd.  2far. 

Analysis. — Write  the  numbers,  as  before,  so  that  units  of  the  samo 
order    shall   fall  in  the   same  column.      Beginning 

■with  the  lowest  denomination,  yve  find  the  sum  to  be  operation. 

5  farthings.  But  since  4  farthings  make  a  penny,  we  £  s.d.far. 
set  down  1  farthing,  and  carry  one  penny  to  the  column  14     7  8     3 

of  pence.  The  sum  of  the  pence  then  becomes  1 8,  which       6  18  ^ 2 

is  1  shilling  and  6  pence  over.    Set  down  the  6  pence,  21     6  6     1 
and  carry  the  1  shilling  to  the  column  of  shillings,  the 

sum  of  which  becomes  26  ;  that  is,  1  pound  and  6  shillings.  Setting 
down  the  6  shillings  and  carrying  1  to  the  column  of  pounds,  we  lind 
the  entire  sum  to  be  £21   6s.  6d.  Ifar. 

Hence,  for  the  addition  of  all  numbers, 

I.  Write  the  numbers  so  that  units  of  the  same  value  shall  fall 
in  the  same  column. 

II.  Add  the  tmits  of  the  lowest  denomination,  and  divide  their 
sum  hj  so  many  as  make  one  unit  of  the  denomination  next  higher. 
Set  down  the  remninder  and  carry  the  quotient  to  the  next  higher 
denomination  ;  proceed  in  the  same  manner  through  all  the  denom- 
inations and  set  down  the  entire  sum  of  the  last  column. 


ADDITION. 


37 


PROOF. 

43.  The  proof  of  an  operation,  in  Addition,  consists  in  show- 
ing that  the  answer  contains  as  many  units  as  there  are  in  all 
the  numbers  added.     There  are  three  methods  of  proof. 

I.  Begin  at  the  top  of  the  units  column  and  add,  in  succession, 
all  the  columns  doicnwards.  If  the  two  residts  agree,  the  work  is 
supposed  to  he  right  ;  for,  it  is  not  likely  that  the  same  mistake  will 
have  been  made  in  both  additions. 

ir.  Divide  the  given  numbers  into  parts,  and  add  the  parts 
separately :  then  add  together  the  partial  sums  ;  if  the  results 
agree,  the  toork  is  supposed  to  he  right  ;  for,  a  whole  is  equal  to  the 
sum  of  all  its  parts. 

III.  Find  the  excess  o/9's  in  each  number,  and  place  it  at  the 
right  (Art.  20).  Add  these  nuinbers  and  note  the  excess  ofd's 
in  their  sum.  This  excess  shoidd  be  equal  to  the  excess  of  9's  in 
the  sum  of  the  numbers. 


1st.  Method. 

2d. 

Method. 

182796 

182796 

32160 

143274 

143274 

47047 

32160 

Partial  sums  326070 

79207 

47047 

Sum  405277 

326070 

1st 

partial  sum. 

79207 

2d 

a 

Sum    405277 

3d  Method. 

182' 

796 

6  excess 

of  9's, 

143: 

274 

3       " 

a 

32: 

160 

3       " 

i( 

47( 

347 

•7 

4       « 

u 

Sura  405277  •• 

16    -     - 

• 

7  excess  of  9'h 

READING. 

44.  The  pupil  should  be  early  taught  to  omit  the  intermediate 


43.  What   is   the    proof  of  an  operation  in   Addition  1 
methods  of  proof  are  there  I     Explain  each  separately  1 

44.  What  18  Ihe  reading  process,  in  Addition"! 


How   many 


38 


ADDiTION. 


words  in  the  addition  of  columns  of  figures.  Tiius,  in  the 
above  examjjle,  instead  of  saying  7  and  0  are  7  ;  7  and  4  are 
eleven  ;  11  and  6  are  seventeen ;  he  should  simply  say,  seven, 
eleven,  seventeen.  Then,  in  the  column  of  tens  he  should  say, 
five,  eleven,  eighteen,  twenty-seven  ;  and  similarly,  for  the 
other  columns  at  the  left.  This  is  called  reading  the  columns. 
Let  the  pupils  be  often  practised  in  the  readings,  both  separately 
and  in  concert  in  the  class. 

EXAMPLES. 


(!•) 

(2.) 

(3.) 

(4.) 

94201 

80032 

98800 

10304 

4G390 

4291 

10926 

67491 

37467 

2376 

321 

1324 

4572 

840 

479 

46 

5.  What  is  the  sum  of 

6.  What  is  the  sum  of 
169? 

7.  What  is  the  sum  of  42300,  6000,  347001,  525,  47 


1376,  38940,  8471,  23607,  891  ? 
3480902,  3271,  567321,  91243,  6001, 


(8.) 

days. 
1276 
3718 
9024 
6357 
1028 
9131 

(13.) 
miles. 
1600 
2588 
9101 
6793 
8267 
4572 


(9.) 

husliels. 

47917 

12031 

5672 

8321 

728' 

4^ 

(14.)  " 
furlongs. 
47468 
59012 
23419 
15760 
27900 
12317 


(10.) 

rods. 
9003 
1881 
6035 
7810 
3176 
2004 

(15.) 
pounds. 
76389 
1036 
2671 
5132 
6784 
1672 


(11.) 

minutes. 

67321 

4702 

1067 

456 

377 

99_ 

(16.) 
dollars. 
1602 
9614 
4732 
5675 
8211 
4455 


(12.) 

gallons. 

760324 

18720 

5762 


359 


1082 
47269 

(17.) 

casks. 

40506 

37219 

50170 

32614 

73462 

10001 


ADDITION. 

•6Si 

(18.) 

(19.)        (20 

.)      (21.) 

(22.) 

$175,3G5   $30,365   $180,000   $300,40 

$4802,279 

278,050 

28,779    489,007    167,275 

1642,107 

420,96 

10,101     76,119     18,197 

3026,267 

76,125 

9,08     16,423    29,94 

125,092 

41,04 

7,14      9,011     10,08 
(24.)        (25.) 

42,75 

(23.) 

(26.) 

£  s.  d.  far 

lb.    oz.  dwt. 

-ft  5  3 

lb.  oz.  dr. 

14  11  3  1 

174  11  19 

17  11  7 

17  15  12 

17  18  10  2 

75  10  13 

94  10  6 

29  32  10 

29  7  6 

642  3  10 

60  9  2 

84  10  9 

42  14  11  3 

125  7  5 

42  3  9 

14  3  7 

17  10  00  1 

62  0  16 

12  0  6 

40  9  9 

84  00  1  0 

39  1  4 

98  7  5 

76  4  7 

16  19  8  2 

176  10  15 

127  1  0 

18  11  15 

(27.) 

(28.) 

(29.) 

(30.) 

ciot.  qr.    lb. 

yd.    qr.  na. 

^.  E.  qr.  na. 

L.  mi.  fur 

174  2  20 

74  3  3 

14  4  3 

n   2   1 

320  1  14 

60  1  2 

75  1  2 

10  1  4 

136  3  23 

14  0  1 

84  3  1 

7  0  6 

47  0  12 

45  2  3 

17  2  0 

5  2  3 

84  1  24 

69  1  0 

10  0  2 

25  1  0 

90  2   9 

11  0  0 

19  1  1 

36  2  2 

7  3   5 

36  3  1 

29  3  2 

40  1  0 

(31.) 

(32.) 

33.) 

(34.) 

yrds.    ft.    in. 

A.     JR.   F. 

Tun.  hlid.  gal. 

gal.  qt.  pt. 

174  11  1 

77  3  39 

714  3  56 

14  3  1 

260   2  0 

64  2  37 

626  1  48 

74  2  1 

150  10  2 

16  1  29 

320  0  29 

96  1  0 

126   9  1 

72  0  18 

156  2  31 

47  2  1 

96   7  0 

36  2  20 

225  1  42 

22  0  1 

72   4  1 

42  2  14 

84  0  17 

65  1  0 

8   6  2 

11  3   7 

96  1  34 

19  0  0 

40 

additio:n. 

(35) 

t 

(36.) 

(37.) 

(38.) 

chal.  hiL 

qt. 

yr. 

mo. 

.  wic. 

da. 

hr. 

min. 

qr.     lb. 

oz. 

14  31 

G 

127 

9 

2 

140 

12 

27 

44  21 

14 

25  14 

2 

320 

10 

3 

340 

16 

40 

14  16 

12 

36  29 

7 

146 

8 

1 

227 

20 

56 

22  10 

n 

42  24 

3 

75 

6 

0 

102 

13 

25 

36  19 

7 

39  32 

1 

70 

11 

2 

67 

21 

37 

51  13 

9 

56  19 

5 

54 

7 

1 

14 

9 

10 

30  22 

11 

14  20 

4 

27 

4 

3 

10 

19 

46 

16  15 

15 

39.  The  population  of  tlie  United  States  and  territories,  in 
1850,  was  as  follows :  White  population,  19553068;  Free 
Colored  population,  434495  ;  Slave  population,  3204313  ;  In- 
dians, 400674  ;  -what  Avas  the  entire  population  ? 

40.  In  the  year  1850,  the  expenditures  of  the  United  States, 
amounted  to  43002168  dollars;  in  1851,  to  48905879  dollars; 
in  1852,  to  46007893  dollars:  what  were  the  expenditures  of 
the  United  States  for  these  three  years  ? 

41.  A  man  of  fortune  bequeathed  to  each  of  his  three  sons, 
10492  dollars  ;to  each  of  his  twodaughters,  5976  dollars  ;  to  his 
wife,  the  remainder  of  his  property,  which  exceeded  the  amount 
bequeathed  to  his  children  by  twelve  hundred  dollars :  find  the 
amount  of  his  property  ? 

42.  A  stage  goes  in  one  day  27  miles  3  furlongs  36  rods; 
the  next,  32  miles  10  rods ;  the  next,  36  miles  2  furlongs ;  the 
next,  25  miles  6  furlongs  38  feet :  how  far  did  it  go  in  four 
days? 

43.  The  population  of  Boston,  in  1854,  was  178000  ;  Provi- 
dence, 60,000  ;  Buflalo,  75000 ;  New  Orleans,  139190;  Louis- 
ville, 55000  ;  Sacramento,  12000':  what  was  the  entii'e  popula- 
tion of  these  cities  ? 

44.  The  population  of  New  York  city,  in  1850,  Avas  515547; 
Brooklyn,  127618;  Baltimore,  169054;  Washington,  40000: 
Cincinnati,  115436;  Chicago,  29963;  St.  Louis,  77860;  Mil- 
waukie,  20061  ;  Detroit,  21119  ;  Indianapolis,  8091  :  what  wu." 
the  entire  population  of  these  cities  ? 


ADDITION.  4:1 

45.  Bouglit  a  barrel  of  flour  for  eight  dollars  and  seventy- 
five  cents ;  a  ton  of  plaster  for  five  dollars  sixty-two  and  a-half 
cents ;  a  hat  for  three  dollars  twelve  cents  and  five  mills  ;  fifty 
pounds  of  sugar,  for  four  dollars  fifty  cents  and  nine  mills;, 
what  was  the  amount  of  my  bill  ? 

46.  A  lady  bought  a  bonnet  for  $5,375  ;  some  silk  for  $12,03  ; 
some  ribbon  for  $0,875  ;  a  shawl  for  $9,46  :  what  did  the  whole 
amount  to  ? 

47.  A  wine-merchant  taking  an  invoice  of  his  liquors,  finds 
that  he  has  ohhds.  36  gah.  2qts.  of  wine  ;  3hhds.  \6fjals.  Iqt.  Ipt. 
of  rum  ;  Ihhd.  2qts.  of  gin  ;  40^a/s.  Ipf.  of  whisky :  how  much 
liquor  in  all  ? 

48.  Tea  Avas  imported  into  the  United  States  in  the  year 
1851,  to  the  value  of  $4798005  ;  in  1852,  17285817  ;  in  1853, 
i^8224853 :  what  was  the  value  of  the  tea  imported  during 
these  three  years  ? 

49.  The  United  States  exported  tobacco,  in  the  year  1851, 
to  the  amount  of  $9219251;  in  1852,  $10031283;  in  1853, 
$11319319  \  what  was  the  entire  value  of  tobacco  exported  in 
these  three  years  ? 

50.  A  man  sold  his  house  and  lot  for  $25840,  w^iich  was 
$3186  less  than  he  gave  for  them ;  how  much  did  they  cost 
him  ? 

51.  A  speculator  bought  three  city  lots,  for  the  first  lie  paid 
$2870,43  ;  for  the  second,  $2346,75;  for  the  third,  $1563,82. 
He  sold  the  same  at  an  average  profit  upon  each  of  $476,25; 
what  amount  did  he  receive  for  the  lots  ? 

52.  The  population  of  England,  in  1851,  was  16921888;  of 
Ireland,  6515794;  of  Scotland,  2888742:  what  was  the  entire 
population  of  the  three  ? 

53.  The  churches  of  the  United  States  and  territories  in 
1850,  were,  Baptists,  9375  ;  Congregationalists,  1706  ;  Presby- 
terians, 4824  ;  Methodists,  13280  ;  Universalists,  529  :  what  was 
the  whole  number  of  churches  belonging  to  these  five  denomi- 
nations ? 

54.  In  the   same  year,  the  value  of  the    church  property 


42  ADDITION. 

owned  by  the  Baptists  in  the  United  States  and  territories, 
was  $]  1020855  ;  by  the  Congregationalists,  §7970195;  by  the 
Tresbyterians,  $14543789;  by  the  Methodists,  $14822870;  by 
the  Universalists,  $1752310:  what  was  the  Avhole  amount? 

55.  During  the  year  1853,  there  was  coined  in  the  United 
States,  $51888882  of  gold;  87852571  of  silver;  and  $07059 
of  copper :  what  was  the  whole  amount  of  money  coined  in  the 
United  States  in  1853? 

56.  A  farmer  sends  to  market  the  following  quantities  of 
butter;  IScwt.  2qrs.  IGlbs. ;  lion  5mot.  21lbs. ;  2qrs,  lAIbs. ; 
how  much  did  he  send  in  all  ? 

57.  A  man  having  84  acres  3  roods  20  rods  of  land,  buys 
120  acres  14  rods  more  :  how  much  did  he  then  have? 

58.  Suppose  a  father  divides  his  estate  equally  among  his 
three  sons,  giving  each  twenty-five  thousand  dollars  seven 
dimes  six  cents  and  five  mills :  what  was  the  value  of  the 
estate  ? 

59.  A  farmer  has  three  fields  of  grain,  the  first  yields  1375 
bushels;  the  second,  1810  bushels;  the  third,  1205  bushels; 
he  values  his  entire  farm  at  $2975  more  than  the  number  of 
bushels  of  grain  raised  from  these  three  fields :  what  was  the 
value  of  his  farm  ? 

60.  Bought  a  silver  tea-pot  weighing  lib.  Goz.  12du't. ;  a 
cream-cup,  weighing  lOoz.  IMict.  20gr.  ;  a  porringer,  weigliing 
Woz.  lOyr. ;  a  dozen  large  spoons,  weighing  l^i^.  l-idwt.  12gr.: 
what  was  the  weight  of  the  whole  ? 

61.  The  whole  number  of  ailults  in  the  United  States  and 
territories,  over  twenty  years  of  age,  who  could  not  read  and 
write  in  1850,  were  as  follows;  of  whites,  males,  3896G4; 
females,  573234  ;  free  colored,  males,  40722  ;  females,  49800  : 
what  was  the  whole  number  ? 

62.  The  whole  number  attending  school  the  same  year,  were, 
of  wliites,  males,  2146432;  females,  1916614;  free  colored, 
males,  13864  ;  females,  12597  :  what  was  the  whole  number 
attending  school  ? 

63.  A   forwarding  merchant  had   in   his  store-room,  at  one 


ADDITION.  43 

dme,  7500  bushels  of  corn  ;  12865  bushels  of  wheat ;  4G80 
bushels  of  oats  ;  329G  bushels  of  barley,  and  had  room  enough 
left  to  store  4000  bushels  of  oats :  how  many  bushels  of  grain 
would  the  storehouse  hold  ? 

64.  A  man  engagmg  in  trade  had  $5164,50,  in  cash; 
$11810,25,  in  goods  ;  $3004,  in  notes.  His  nett  profits  aver- 
aged $2384,16,  a  year,  for  3  years  :  what  was  the  total  value 
of  the  property  at  the  end  of  the  three  years  ? 

65.  A  person  paid  two  eagles  for  a  coat ;  four  dollars  and 
six  dimes  for  a  hat ;  two  dollars  and  sixty-three  cents  for  a  vest; 
ei^-ht  dimes,  seven  cents  and  five  mills  for  a  knife  :  what  was 
the  amount  of  his  bill  ? 

66.  From  a  piece  of  cloth,  I2ijds.  2qrs.  were  cut  at  one  time  ; 
16)jds.  Iqr.  3na.  at  another,  Avhen  there  were  lOyds.  Iqr.  Ina. 
remaining  :  how  much  was  there  in  the  whole  piece  ? 

67.  A  farmer  purchased  a  plough  for  $9^ ;  a  wagon,  for 
$45i- ;  a  horse,  for  $110f  ;  a  load  of  hay,  for  $121;  a  harrow, 
for  $34- :  what  was  the  cost  of  the  whole  ? 

68.  If  a  certain  warehouse  be  worth  $12540,371,  and  one- 
fourth  the  contents  is  valued  at  $5632,108  :  what  is  the  value 
of  the  warehouse  and  the  whole  of  its  contents  ? 

60.  The  number  of  bales  of  cotton  used  in  manufactures  in 
1850,  by  Massachusetts,  were  223607  ;  New  Hampshire,  83026  ; 
Ehode  "^Island,  50713;  Virginia,  17785;  New  York,  37778; 
Maryland,  23525  ;  Connecticut,  39483  ;  Alabama,  5208  :  what 
was  the  entire  amount  consumed  in  those  states  ? 

70.  In  1850,  the  State  of  New  York  produced  13121498 
bushels  of  wheat ;  Pennsylvania,  15367691  bushels;  Virginia, 
11212616  bushels  ;  Ohio,  14487351  bushels  ;  Missouri,  2981652 
bushels  ;  Illinois,  9414575  bushels:  what  was  the  whole  num- 
ber of  bushels  produced  by  those  states  in  that  year  ?    . 

71.  A  farmer  sold  his  wheat  for  $825,87^-;  his  barley  for 
67,121;  liis  pork  for  $80,10;  his  apples  for  $46:  how  much 
did  he  receive  for  the  whole  ? 

72.  Three  persons  enter  into  copartnership,  the  first  put  in 
7825  dollars  capital ;  the  second  put  in  1250  dollars  more  than 


44  SUBTRACTION. 

the  first ;  and  the  third  put  in  as  much  as  the  other  two :  what 
was  the  whole  amount  of  capital  invested  ? 

73.  A  farmer  raised  in  one  field  2A0bnsIi.  3ph.  2qts.  of 
wheat ;  in  another  97bush.  Gqts. ;  in  another  A2hush.  IpJc. :  how 
much  did  he  raise  in  the  three  fields  ? 

74.  Add  together  three  hundred  dollars,  ten  eagles,  forty 
dimes,  ninety-six  cents,  seven  mills,  nine  dollars,  forty-seven 
cents,  five  mills,  four  eagles,  three  dollars,  and  nine  dimes. 

75.  What  is  the  sum  of  £17  10s.  6d.;  £25  4s.  lO^d. ;  18s. 
Gd.  3far.;  £11  Q^^d.;  £1  18s.;  21s.  O^d.? 

76.  The  Deluge,  according  to  Chronology,  occurred  1656 
years  after  the  creation ;  the  call  of  Abraham  427  after  the 
Deluge  ;  the  departure  of  the  Israelites,  430  after  the  call  of 
Abraham ;  the  foundation  of  the  temple,  479  after  the  de- 
parture of  the  Israelites  ;  the  end  of  the  captivity,  476  after 
the  foundation  of  the  temple  ;  and  the  birth  of  Christ,  536  years 
after  the  end  of  the  captivity :  how  many  years  from  tlie  crea- 
tion to  the  present  time,  it  being  the  year  1856  ? 

SUBTRACTION. 

45.  The  Difference  hetween  two  numbers  is  such  a  number 
as,  added  to  the  less,  will  give  the  greater. 

If  the  numbers  are  unequal,  the  larger  is  called  the  minuend, 
and  the  less  the  subtrahend.  If  they  are  equal,  either  is  the 
minuend  and  the  other  the  subtrahend.  Their  difference, 
whether  they  are  equal  or  unequal,  is  called  the  remainder. 

Subtraction  is  the  operation  of  finding  the  difference  he- 
tween two  numbers. 

1.  From  869  take  327  ;  that  is,  from  8  lumdreds  6  tens  and 
9  units,  take  3  hundreds  2  tens  and  7  units. 

45.  "What  is  the  ililTerPiico  Ix-twcen  two  numbers  !  When  the  minibera 
are  unequal,  what  is  the  larijcr  nuiiiher  callcil !  "VViiat  is  the  h^ss  eallcd  1 
Wliat  is  their  ditlcrnnce  called  1  ^^'hat  is  Subtraction  ^  Give  tlie  rule 
for  findini'  the  diflercnce  between  two  numbers. 


OPERATION. 

TT 
C 

2 

GO 

10 

624    - 

5 

12 

4 

393    - 

3 

9 

3 

1 

SUBTRACTION.  45 

Analysis. — Beginning  -wilh  the  right  .land         operation. 

rgure,  we  take  units  from  units  ;    then,  tens  869  minuend, 

from  tens  ;  then,  hundreds  from  hundreds,  and  327  subtrahend, 

find  tlie  remainder  to  be  542.  542  remainder. 

2.  From  624  take  393. 

Analysis. — Having  Avritten  do\vn  the  numbers,  Ave  subtract  3  fron? 
4,  and   find  a  remainder  1.     At  the  next  step 
Ave  meet   a  difTiculty,  for  we   cannot  subtract 
9  tens  from  2  tens. 

Take  1  hundred  =  10  tens,  from  6  hun- 
dreds and  add  it  to  2  tens.  Then  9  tens  from 
12  lens  leaves  3  tens,  and  3  liuiidreds  from  5 
leaves  2  hundreds,  and  the  remainder  is  231.         231-2       3     1 

The    remainder    can   be    found  by  adding, 
mentally,  10  to  2  tens,  and   then  saying,  9  from  12   leaves  3  tens; 
then  adding  1  to  3  hundreds,  and  say,  4  from  6  leaves  2  hundreds. 

The  process  of  adding  10  to  a  figure  of  the  minuend  and  returning 
1  to  the  next  figure  of  the  subtrahend,  at  the  left,  is  called  borrowing. 

3.  From  GT.  licwt.  2qr.  20lb.  Uoz.,  take  4T.  ITcio..  \qr. 

2\Ib.  IOgz. 

Analysis. — Taking  \Ooz.  from  1203.,  loz. 
remain.  At  the  next  step  we  find  a  difllcuUy, 
for  2\lb.  cannot  betaken  for  20/6.  "We  then 
take  \qr.  =  25/6.  from  iqr.  and  add  it  to  20/6., 
making  45/6. ;  then  say,  21/6.  from  45/6.  leaves 
24/6.;  we  then  add  1  to  the  next  left  hand 
figure  of  the  subtrahend,  and  .«ay  %qr.  from 
2qr.  leaves  0:  then  llcict.  from  'iicici.  leaves 
\1cict ,  and  5  from  6  leaves  1  ton. 

Hence,  to  find  the  difference  between  two  numbers, 

I.  Set  down  the  less  nimiber  under  the  greater,  so  thai  units  of 
the  same  value  shall  fall  in  the  same  column. 

II.  Begin  with  the  units  of  the  lowest  denomination  and  sub- 
tract each  number  f-om  the  one  above  it. 

III.  When  the  units  of  any  denomination  in  the   subtrahend 
exceed  those  of  the  same  denomination  in  the  yninuend,  suppose 


T. 

cui. 

20 

qr 

lb. 

oz. 

6 

14 

2 

SO 

12 

4 

17 

1 

21 

10 

1 

] 

1 

17 

0 

24 

2 

5 

34 

1 

45 

12 

4 

17 

! 

21 

10 

,  46  SUBTRACTION. 

SO  mamj  units  to  he  added  as  make  one  unit  of  the  next  MgJier 
denomination  ;  after  which  add  1  to  the  next  denomination  of 
the  subtrahend,  and  subtract  as  before. 

FIRST    PROOF. 

46.  The  difference  or  remainder,  is  such  a  number  as  added 
to  the  subtraliend,  will  give  a  sum  equal  to  the  minuend  (-45)  ; 
hence, 

Add  the  remainder  to  the  subtrahend.  If  the  worh  is  right, 
the  sum  zvill  be  equal  to  the  minuend. 

SECOND    PROOF. 

Since  the  remainder  added  to  the  subtrahend  is  equal  to  the 
minuend  (Art.  45),  it  follows  that  the  excess  of  9's  in  these  two 
numbers  is  equal  to  the  excess  of  9's  in  the  minuend  ;  hence, 

Find  the  excess  of  9's  in  the  minuend,  in  the  subtrahend  and 
in  the  remainder  ;  if  the  work  is  right,  the  excess  of  9's  tJi  the 
two  last  numbers  will  be  equal  to  the  excess  ofd's  in  the  first. 

What  is  the  difference  between  874136  and  45302  ? 

2d.  Method. 

-  2  excess  of  9's  in  the  first. 

-  5  «         «  2d. 
_                                -G+5  =  ll:  2  excess. 

READING. 

47.  What  is  the  difference  between  426  and  295  ? 

By  the  common  method,  wliieh   is  spelling,         operation. 
wo  say,  5  from  6  leaves  1;  9  from   12  leaves  426 

3  ;   1  to  carry  to  2  are  3  ;  3  from  4  leaves  1.  295 

By  reading   the  words   which   express  the  131 

final  result,  we  make  the  operations  mentally,  and  say,  o?Jc.  three, 
cnc. 


4G.   \Miat  is  the  diflcrcnce  or  ri'maitider  !     How  do  you  i)iove  Sulttrac- 
tion  ''     How  do  you  prove  Subtraction  by  the  second  method  ! 
47.  Explain  the  process  of  reading  in  Subtraction. 


1st.  Method. 

874136 

874136 

45302-1 

45302 

828834 

828834 

SUBTRACTION. 


47 


OPERATION. 

yr. 

mo. 

da. 

h.r. 

1855 

1 

4 

15 

1801 

3 

4 

12 

TIME    BETWEEN    DATES. 

48.  "What  time  elapsed  between  the  inauguration  of  INIr. 
JefFerson,  March  4th,  12  o'clock,  M.,  1801,  and  July  4tli,  3 
P.  M.,  1855  ? 

Analysis. — Place  the  earlier  date  under  the 
later,  writing  the  number  of  tlie  year,  reckoned 
from  the  beginning  of  the  Christian  Era,  on  the 
left.     Then,  write  in  the  same  line  the  num- 
ber of  the  month,  reckoned   from   the  first  of  54     4     0       3 
January,  the  number  of  tlie  day,  reckoned  from  tlie  first  of  the  month, 
tlie  number  of  the  hour,  reckoned   from  12   at  night,  and  write   tliei 
number  of  minutes   and  seconds,  if  there  are  any,  still  at  the  right. 
Hence,  to  find  the  time  between  two  dates, 

Write  down  as  above,  and  subtract  the  earlier  date  from  the  latter. 

Note  1 . — In  finding  the  difference  between  dates,  as  in  casting 
interest,  the  month  is  regarded  as  tlie  twelfth  part  of  the  year  and 
as  containing  30  days. 

2.  The  civil  day  begins  and  ends  at  12  o'clock  at  night. 


EXAMPLES. 

(1.) 

(2.) 

(3.) 

(4.) 

From 

472567 

103796 

900372 

1760134 

Take 

.  109271 

47217 

167301 

48207 

(5.) 

(6.) 

(7.) 

(8.) 

rods. 

dollars. 

mills. 

barrels. 

From 

74623457 

8600000 

162347 

8462 

Take 

32700169 

761820 

56321 

4071 

(10.) 

(11.) 

bushels. 

inches. 

minutes. 

From 

100000 

200763194 

3 

601789412 

Take 

37214 

2142079 

— 

10031761 

48.  How  do  you  find  the  difference  of  time  between  two  dates!  In 
this  computation,  what  part  of  a  year  is  a  month  \  How  many  days  arc 
rccko-ned  to  the  month  1 


48 


6 

SUBTRACTION. 

(12.) 

(13.) 

(14.) 

cords. 

gallons. 

pounds. 

From 

4200000 

^^ii^in 

100000000 

Take 

325 

9999 
(16.) 

23 

(15.) 

(17.) 

From 

$8475,656 

$1000,759 

$4871036,008 

Take 

32,015 

194,375 

17362,25 

(18.) 

(19.) 

(20.) 

£       s.     d.  far 

.     ton.  cwt.  qrs.  lb. 

yd.    qr.  ncu 

From 

25     12     6     2 

5     17     3     21 

137     1     3 

Take 

10     14     3     1 

2       9     1     14 

19     3     2 

(21.) 

(22.) 

(23.) 

X.  mi.  fur.  rd. 

tun.  hhd.  gal.  qt.  pt. 

A.   R.  P. 

From 

16    2    1    2,1 

14    1     26    2     1 

100    2    27 

Take 

16     1     4      9 

5    3    35    3    1 

10    3    30 

24.) 

(25.) 

(26.) 

hush.   ph.  qt. 

cords,   ft.      in. 

E.E.  qr.  na. 

From 

1000     3     4 

225     42     1242 

42     1     2 

Take 

25     1     6 

100  112       720 

16     4     3 

(27.)  (28.)  (29.)  (30.) 

ft   3   3  3   3   9  E.E  qr.  na.  E:F.    qr.  na. 

144  10  5  27  4  1  174  3  1  171  1  3 

64  11  7  14  7  2  49  4  2  74  3  2 


(31.) 

(32.) 

(33.) 

(34.) 

2\     civt.  qr. 

cwt.    qr.    lb. 

qr. 

lb.     oz. 

lb.     oz.    dr. 

14     12     2 

17     1     21 

143 

22     12 

174     11     10 

1     14     3 

14     2     24 

74 

19     14 

39     12     13 

(35.) 

(36.) 

(37.) 

(38.) 

A.  R.  P. 

A.    R.  P. 

da. 

hr.    min. 

Jir.    min.  sec. 

12    1    32 

112    1    31 

167 

21     50 

147     50     51 

1    3    14 

74    2    37 

19 

23     54 

94     59     57 

SUBTRACTION.  49 

39.  From  $10000  take  $1240,37i-. 

40.  From  183701289  take  34627. 

41.  From  17^/-.  9mo.  Iwk.  \Ma.  take  lOy.  llmo.  2wk.  bda. 

42.  From  1441fe  7  3   5  3   19  take  56t5  6  !   7  3   19. 

43.  From  two  eagles  seven  dimes,  take  twelve  dollars  and 
fifty  cents. 

44.  From  forty  dollars  twelve  and  a  half  cents,  take  twenty- 
five  cents  and  seven  mills. 

45.  From  one  eagle  five  dollars  six  dimes  and  ten  cents, 
take  five  dollars  seven  cents  and  four  mills. 

46.  What  sum  added  to  £11  145.  'd\d.  will  make  £133  11*. 
^dl 

47.  An  apprentice,  who  is  14  years  11  months  3  weeks, 
14  hours  58  minutes  old,  is  to  serve  his  master  until  he  is  21 
years  of  age.     How  long  has  he  to  serve  ? 

48.  Tiie  greater  of  two  numbers  is  seven  millions  three 
hundred  and  four  thousand  and  ten  ;  the  less  is  nine  hundred 
and  fifty  thousand  one  hundred  and  forty.  What  is  their  differ- 
ence ? 

49.  Mont  Blanc,  the  highest  mountain  in  Europe,  is  15680 
feet  high  ;  Chimborazo,  the  highest  in  America,  is  21427  feet. 
What  is  the  difierence  in  their  heights  ? 

50.  A  man  sold  his  farm  for  seven  thousand  five  hundred 
and  thirty  dollars,  which  was  fifteen  hundred  and  ten  dollars 
more  than  he  gave  for  it.     How  much  did  he  give  for  it  ? 

51.  The  revenue  collected  at  the  port  of  New  York  for  the 
year  ending  30th  June,  1853,  was  $38289341,58  ;  at  Philadel- 
phia, $4537046,16  ;  at  Boston,  $7203048,52  ;  at  Baltimore, 
$836437,99.  How  much  more  was  collected  at  the  port  of 
New  York  than  at  the  other  three  ? 

52.  A  man  engaging  in  trade  found  at  the  end  of  five  years 
that  he  had  increased  his  capital  ten  thousand  three  hundred 
and  ten  dollars,  and  that  his  whole  capital  amounted  to  forty- 
six  thousand  five  hundred  dollars.  How  much  did  he  com- 
mence with  ? 

3 


60  SUBTRACTION. 

53.  The  minuend  exceeds  the  remainder  by  683021,  and  the 
remainder  is  902563.     What  is  the  subtrahend  ? 

54.  The  amount  of  tea  consumed  in  the  United  States  in  the 
year  1846,  was  16891020  pounds;  the  amount  of  coffee. 
124336054  pounds.  How  much  more  coffee  than  tea  ^Yas 
consumed  ? 

55.  "What  number  is  that  to  which,  if  you  add  3726,  the  sum 
will  be  ten  thousand  ? 

56.  From  a  stack  of  hay  containing  dtons  Sgr.  20Ib.,  I  sold 
4:tons  17 cwt.  22lb.     How  much  was  then  left  ? 

57.  A  owes  B  £25  ;  after  paying  him  £5  9ic?.,  how  much 
will  he  still  owe  him  ? 

58.  If  the  distance  from  New  York  to  Liverpool  be  3100 
miles,  after  a  ship  has  sailed  800/rt.  ofur.  36roc?s,  what  distance 
remains  ? 

59.  Bought  a  farm  for  three  thousand  five  hundi'ed  dollars 
and  fifty  cents  ;  sold  the  same  for  three  thousand  three  hundred 
dollars  and  eighty-seven  and  a  half  cents  :  how  much  did  ho 
lose  by  the  bargain  ? 

60.  If  a  lot  of  goods  are  bought  for  $750,  and  sold  for 
$925,871  what  will  be  gained  ? 

61.  If  I  buy  a  bushel  of  wheat  for  $1,871 ;  ten  gallons  of 
molasses  for  $2,50  ;  five  yards  of  cloth  for  $12,371 :  how  much 
chanjre  must  I  receive  back  for  two  ten  dollar  bills  ? 

62.  The  population  of  the  United  States  in  the  year  1850 
was  23191876,  of  which  3204313  were  slaves:  what  was  the 
white  population  ? 

63.  England  contains  50922  square  miles ;  Scotland,  31324 
square  miles  ;  "Wales  7398  square  miles ;  the  United  States 
contain  2988892  square  miles.  How  many  more  square  miles 
does  the  United  States  contain  than  the  whole  of  Great 
Britain  ? 

64.  A  gentleman  of  fortune  owning  an  estate  of  two  hun- 
dred thousand  dollars,  bequeathed  thirty  thousand  dollars  to 
objects  of  chaj'ity  ;  twenty-five  thousand  two  lunulrcd  and  fifty 
dollars  to  each  of  his  three  sons;  twenty  thousand  five  hundred 


SUBTRACTION.  51 

and  seventy-five  dollars  to  his  daughter ;  and  the  remainder  to 
his  widow.     How  much  did  the  widow  receive  ? 

65.  The  population  of  New  Orleans  in  1850  was  116375; 
in  1854  it  was  139190  :  what  was  the  increase  in  four  years  ? 

66.  Having  deposited  $1500  in  a  bank,  I  drew  out  at  one 
time  $475,121;  at  another  time  $300;  at  another  $526,25: 
how  much  remained  ? 

67.  If  the  Declaration  of  Independence  was  made  at  pre- 
cisely 12  o'clock,  on  the  4th  day  of  July,  1776  ;  how  much  time 
will  have  passed  to  the  4th  day  of  March,  1857,  at  30  minutes 
past  3  o'clock,  P.  M.  ? 

68.  If  I  borrow  $1576  of  a  friend,  and  afterwards  pay  him 
$920,871,  how  much  would  I  still  owe  him  ? 

69.  The  first  settlement  made  in  the  United  States  was  at 
Jamestown,  in  Virginia,  May  23,  1607  :  how  many  years  from 
that  time  to  the  4th  of  July,  1856  ? 

70.  The  sum  of  two  numbers  is  36804,  and  the  less  number 
is  eighteen  thousand  nine  hundred  and  twenty-seven  :  what  is 
the  sri'eater  number  ? 

71.  The  revenue  of  the  United  States  in  the  year  1853  was 
$61337574 ;  the  expenditures  $54026818  :  how  much  did  the 
revenue  exceed  the  expenditures  ? 

72.  Tlie  exports  of  the  State  of  New  York  in  the  year  1853 
were  valued  at  $66030355  ;  those  of  the  State  of  Virginia,  for 
the  same  year,  $3302561  :  how  much  did  the  exports  of  New 
York  exceed  those  of  Virginia  ? 

73.  A  ship-builder  sold  a  vessel  for  $50376,  which  cost  him 
$42978  :  how  much  did  he  gain  ? 

7  I.  A  farmer  sold  his  farm  for  six  thousand  three  hundred 
and  seventy-five  dollars  ;  after  paying  his  debts,  he  has  four 
thousand  and  fifteen  dollars  left :  Avhat  was  the  amount  of  his 
debts  ? 

75.  Gunpowder  was  invented  in  the  year  1330  :  how  many 
years  from  that  time  to  the  year  1856  ? 

76.  What  number  is  that  to  which  if  you  add  3726  the  sum 
will  be  ten  thousand  ? 


SUBTKACTION. 

77.  A  speculator  bought  a  quantity  of  flour  for  $2084,50  ; 
of  bacon  for  $760,871 ;  of  hops  for  $1836,25.  He  sold  the 
flour  for  §2375,60;  the  bacon  for  $912,375;  the  hops  for 
$1750 :  what  did  he  gain  or  lose  on  the  \Yhole  ? 

78.  A  farmer  has  two  pastures,  one  containing  9A.  3Ii.  32P. ; 
the  other  12^.  29P.  He  has  also  two  meadows,  one  contain- 
ing 10^.  2Ii. ;  the  other  15^,  IB.  20P. :  how  much  more 
meadow  than  pasture  has  he  ? 

79.  From  a  pile  of  wood  containing  76  cords  and  6  cord 
feet,  was  taken  at  one  time  20  cords  and  48  cubic  feet ;  at 
another  time  14  cords  1  cord  foot  and  80  cubic  feet :  how 
much  remained  in  the  pile  ? 

80.  A  gentleman  purchased  a  house  wortli  $9430  ;  a  cai*- 
riage  for  $475,50 ;  a  span  of  horses  for  $840,40.  He  paid  at 
onetime  $5260;  at  another  $1275,371;  at  another  $936,42: 
how  mucli  remained  unpaid  ? 

81.  If  a  ship  and  cargo  are  valued  at  $47568,487,  and  the  cargo 
alone  at  $3406,50,  what  is  the  value  of  the  ship  without  the  cargo? 

82.  A  gentleman  dying  left  an  estate  of  $50000  ;  after  pay- 
ing his  debts,  which  amounted  to  $5647,50,  he  desired  that 
each  of  his  two  sons  should  receive  $15000,  and  his  widow  the 
remainder  :  how  much  did  the  widow  receive  ? 

83.  A  note  on  interest,  dated  July  1st,  1853,  was  to  be  paid 
March  20th,  1856  :  how  long  was  it  on  interest? 

84.  Bouglit  a  hogshead  of  wine,  from  Avhich  was  drawn 
S2ffals.  Iqt.  'ipt. ;  how  much  i-emained  in  the  cask  ? 

85.  The  population  of  Chicago  in  1850  was  29963  ;  in  1855 
it  was  80025  :  what  was  the  increase  in  three  years  ? 

86.  A  land  speculator  owning  twenty-five  thousand  acres  of 
land,  sells  at  one  time  fifteen  hundred  acres  ;  at  anotlier  four 
thousand  seven  hundred  ;  at  another  twenty-iive  hundred  acres  ; 
at  anotlier  seven  hundred  and  fifty  acres  :  what  number  of  acres 
has  he  left  ? 

87.  The  latitude  of  New  Orleans  is  29°  57'  30";  tiiat  of 
Boston,  42°  21'  23":  what  is  the  diflerence  in  the  latitude  of 
these  two  places  ?    » 


MULTIPLICATION.  53 

88.  A  person  bought  a  span  of  horses  for  three  hundred 
dollars  ;  a  carriage  for  $410,50 ;  a  harness  for  $50,G7o  ;  he 
sold  the  wliole  for  six  hundred  dollars  :  did  he  gain  or  lose,  and 
how  much  ? 

89.  The  population  of  Great  Britain  and  its  adjacent  islands 
in  the  year  1841,  was  18GG47G1  ;  in  1851  it  was  209364G8  : 
Avhat  was  the  increase  of  population  in  ten  years  ? 

90.  From  a  piece  of  cloth  containing  47  yards,  a  tailor  cut 
I'^yds.  oqrs.  2na.  :  how  much  was  left  ? 

91.  A  ti'adesman  failing  in  business,  was  indebted  to  A 
£105  19s.  lid.;  to  B,  £127  10s.  9i(7. ;  to  C,  £34  I85.  lOd. ; 
to  D,  £500  19s. ;  to  E,  £700  14s.  Gif/.  When  this  took  place, 
he  had  in  cash  £50,  in  goods  to  the  amount  of  £350  14s.  ^d., 
his  household  furniture  was  worth  £24  lis.,  his  book  accounts 
amounted  to  £94  14s.  8c?.  If  all  these  were  given  np  to  the 
creditors,  how  much  would  they  lose  ? 


MULTIPLICATION. 

48.  Multiplication  is  the  operation  of  talcing  one  number 
as  many  times  as  there  are  units  in  another. 

The  number  to  be  taken  is  called  the  midtiplicand. 

The  number  denoting  how  many  times  the  multiplicand  is 
taken,  is  called  the  midtiplier. 

The  result  of  the  operation  is  called  the  product. 

The  multiplicand  and  multiplier  are  called  factors,  or  pro- 
ducers of  the  product. 

Note. — Since,  when  the  multiplier  is  an  integral  number,  the 
product  may  be  obtained  by  adding  the  multiplicand  to  itself  as  many 
times  less  1  as  there  are  unils  in  the  multiplier,  Multiplication  is 
sometimes  called  a  short  method  of  addition. 

48.  What  is  Multiplication  '  What  is  the  numl>er  to  be  taken  called! 
What  docs  the  multiplier  denote  1  What  is  the  result  called  !  What  are 
the  multiplier  and  multiplicand  called  1  Why  is  Multiplication  called  a 
short  method  of  Addition  1 


54  MULTIPLICATION. 

49.  Thei'e  are  three  parts  in  every  operation  of  multiplica* 
tion.  First,  the  multiplicand:  second,  the  muUiplier :  and  third, 
the  product. 

From  the  definition  of  Multiplication,  we  see  that, 

Is;;.  If  the  multiplier  is  1,  the  product  Avill  be  equal  to  the 
multiplicand. 

2c?.  If  the  multiplier  is  greater  than  1,  the  product  will  be  as 
many  times  greater  than  the  multiplicand,  as  the  multiplier  is 
greater  than  1. 

3d.  If  the  multiplier  is  less  than  1,  that  is,  if  it  is  a  proper 
fraction,  then  the  product  will  be  such  a  part  of  the  multipli- 
cand as  the  multiplier  is  of  1. 

50.  Let  it  be  required  to  multiply  any  two  numbers  together, 
say  6  to  4. 

Analysis. — If  we  write,  in  a  horizontal  line,  fi 

as  many  stars  as  there  are  units  in  the  nmlti-  , '^ \ 

plicand,  and  write  as  many  such  lines  as  there 

are  units  in  the  multiplier,  it  is  evident  that, 

4  ■ 
all  the  stars  will  represent  the  number  of  units 

which   arise  from  taking  the  multiplicand   as 

many  times  as  there  are  units  in  the  multiplier. 

Change  now  the  multiplier  into  the  multiplicand  :  that  is,  multi- 
ply 4  by  6. 

Make  in  a  vertical  line,  as  many  stars  as  there  are  units  in  the 
new  multiplicand,  (4),  and  as  many  vertical  lines  are  there  are  units 
in  the  new  multiplier,  (6),  when  it  is  again  evident  that,  all  the  stars 
will  represent  the  number  of  units  in  the  product.     Hence, 

T7ie  product  of  two  factors  is  the  same  whichever  factor  is 
used  as  the  midtipUer. 

3x7  =  7x3  =  21:  also,  Gx3  =  3x6  =  18. 
9x5  =  5x9  =  45:  also,  8x6  =  6x8  =  48. 

and,  8x7  =  7x8  =  56:  also,  5x7  =  7x5  =  35. 

49.  How  many  parts .  are  there  in  every  operation  of  Multiplication  1 
Wliat  are  they  ?  How  many  principles  follow  from  the  definition  of  Mul- 
tiplication ■?     What  arc  they  ! 

50.  In  how  many  way.s  mny  6  and  4  be  multiplied  together  I  How  do 
the  two  produf.ts  compare  with  each  other  1     \\'lial  docs  this  proved 


vf>  ^  9f-  ^  9f  ^ 
•ff-  ^  ^  ^  ^  ^ 
^     ^     *•      ^     ^     ^ 


MULTIPLICATION.  55 

52.  A  Composite  Number  is  one  that  may  be  produced  by 
the  multiplication  of  two  or  more  numbers,  called  factors^ 
Thus,  2  X  3  rz:  6,  in  which  G  is  the  composite  number,  and 
2  and  3  the  factors.  Also,  16  r=:  8  X  2,  in  which  IG  is  a  com- 
posite number,  and  8  and  2  the  factors  ;  and  since  4  X  4  =  16, 
we  may  also  regard  4  and  4  as  factors  of  1 G. 

A  Prime  Number  is  one  which  cannot  be  produced  by  mul- 
tiplication, and  is  divisible  only  by  itself  and  1. 

53.  Let  it  be  required  to  multiply  7  by  the  composite  number 
6,  of  which  the  factors  are  2  and  3. 


7 

_3 

21 

42 


C£> 


7 

« 
* 
* 

* 
* 
* 

* 
* 
* 

* 
* 
* 

* 
* 
* 

* 
* 
* 

*  1 

*  J 
'*  1 

1-2  X  7=  14 

»                    3 

»                 

* 

* 

* 

* 

* 

* 

*  J 

;-2             42 

* 

* 

* 

* 

* 

« 

*  1 

*2 

* 

* 

* 

* 

* 

* 

*  J 

1 

If  we  write  6  horizontal  lines  with  7  units  in  each,  it  is  evi- 
dent that  the  product  of  7  X  6  =  42,  will  express  the  number 
of  units  in  all  the  lines. 

Let  us  first  connect  the  lines  in  sets  of  two  each,  as  at  the 
right ;  the  number  of  units  in  each  set  will  then  be  expressed 
by  7  X  2  =  14.  But  there  are  3  sets;  hence,  the  number  of 
units  in  all  the  sets,  is  14  x  3  =  42. 

Again,  if  we  divide  the  lines  into  sets  of  3  each,  as  at  the 
left,  the  number  of  units  in  each  set  will  be  equal  to  7  x  3  =  21, 
and  since  there  are  two  sets,  the  whole  number  of  units  will  be 
expressed  by  21  X  2  =  42. 

53.  What  is  a  composite  number  1  Give  an  example  of  a  composite 
number  1  What  are  its  factors  1  What  are  the  factors  of  161  What  is 
a  prime  number  1 

53.  If  several  factors  are  multiplied  together,  will  the  product  be  altered 
jy  changing  their  order  1     How  do  you  multiply  by  a  composite  number ' 


56  irULTIPLI'JATlON. 

Since  the  product  of  either  two  of  the  three  factors  7,  3  and 
2,  will  be  the  same  whichever  be  taken  for  the  multiplier  (Art. 
50),  and  since  the  same  principle  will  apply  to  that  product  and 
lo  the  other  foctor,  as  well  as  to  any  additional  factor,  if  intro- 
iuced,  it  follows  that, 

The  product  of  any  number  of  factors  will  he  the  same  in 
vhatever  order  they  are  multiplied : 

Hence,  to  multiply  by  a  composite  number, 

I.  Separate  the  composite  number  into  its  factors : 

II.  Midtiply  the  midtiplicand  and  the  partial  products  hy  the 
factors,  in  succession,  and  the  last  product  will  be  the  entire  pro- 
duct sought. 

Note. — Any  number  whatever,  as  440,  ending  with  0.  is  a  com- 
posite number  of  which  10  is  a  factor:  for,  440  =  44  x  10.  If  there 
are  two  O's  on  the  right  of  the  significant  figures,  then  100  is  a  fac- 
tor, and  so  on  for  a  greater  number  of  ciphers.  Hence,  when  there 
are  ciphers  on  the  right  of  significant  figures,  either  in  the  multipli- 
cand or  multiplier,  or  both, 

Multiply  the  significant  figures  together,  and  then  annex  the  ciphers 
to  the  product. 

54.  1.  Multiply  G27  by  214. 

Analysis. — The  multiplicand  627  is  to  be  ta-  operation. 

ken  214  times :  that  is,  4  units  times,  1  ten  times,  627 

and  2  hundred  times.    Taking  it  4  units  times,  214 

gives  2508 ;   taking  it  1  ten  times  gives  627,  of  2508 

which  the  lowest  unit  is  1  ten  ;  hence.  7  is  written  627 

in  the  tens  place :  taking  it  2  hundred  times,  gives  1 254 

1254.  the  lowest  unit  of  which  is  1   hundred.  134178 
Adding,  we  have  134178  for  the  product. 


Note. — What  is  one  factor  of  a  number  ending  in  0  '  "What  is  one 
factor  of  a  number  ending  in  two  O's  !  In  throe  0"s  !  &c.  How  du  vou 
multiply  by  such  a  number  when  there  are  cipliers  in  one  or  both  factors  ? 

54.  Explain  the  operation  of  niulliplying  6'-i7  by  214.  Explnin  the  five 
principles  which  come  from  this  analysis.  What  is  a  partial  produ -t  ! 
Give  the  general  rule  for  multiplication.  What  must  be  observed  in  tli« 
multiplication  of  United  States  money? 


MULTIPLICATION.  57 

It  is  seen,  from  the  preceding  analysis,  that 

1.  If  units  be  multiplied  hy  units,  the  unit  of  the  product  will 
he  1. 

2.  If  tens  be  multiplied  by  units,  the  unit  of  the  product  will 
he  1  ten. 

3.  If  hundreds  be  multiplied  by  imits,  the  unit  of  the  product 
will  he  1  hundred;  and  so  on: 

And  since  the  product  of  the  factors  is  the  same  whichever 
is  taken  for  the  multiplier  (Art.  50),  it  follows  that, 

4.  If  units  of  the  first  order  he  multiplied  hy  units  of  a  higher 
order,  the  units  of  the  product  will  be  the  same  as  that  of  the 
higher  order. 

5.  If  units  of  any  order  he  multiplied  hy  units  of  any  other 
order,  the  iinit  of  the  product  luill  he  of  an  order  one  less  than 
the  sum  of  the  units  denoting  the  two  orders. 

Note. — When  the  multiplier  contains  more  than  one  figure,  the 
product  obtained  by  multiplying  the  multiplicand  by  a  single  figure, 
is  called  a  •partial  product.  In  the  last  example  there  are  three 
partial  products,  2508,  627,  and  1254.  The  sum  of  the  partial  pro- 
ducts is  equal  to  the  product  sought : 

2.  Multiply  £3  8s.  M.  3/ar.  by  6. 

Analysis. — Multiplying  3  farthings  by  6,  we  operation. 

have  1 8  farthings,  equal  to  Ad.  and  2far.  :  set         £    s.    d.  far. 
down  the  2/ar.  :  then,  6  times  Qd.  are  36rf.,  and  3     8     6     3 

4  pence  to  carry  are  40fZ.,  equal   to  3  shillings  6 

and  Ad.:  then,  6  times  85.  are  485.  and   3s.  to        20  11     4     2 
carry  are  51  shillings,  equal  to  £2  and  11  shil- 
lings: then,  6  times  jC3  are  jClS  and  jC2  to  carry  are  jC20,  which 
set  down. 

Note. — The  vmit  of  each  product  will  be  the  same  as  the  unit  of 
the  multiplicand.  Hence,  for  the  multiplication  of  all  numbers,  we 
have  the  following 

Rule. — Multiply  every  order  of  units  in  the  multiplicand,  in 
succession,  beginning  with  the  lowest,  hy  each  figure  in  the  mul- 
tiplier, and  divide  each  product  so  formed  hy  so  many  units  as 

naJce  one  unit  of  the  next  higher  denomination  :  write  down  each 

3* 


5S  MULTIPLICATION. 

remainder  under  the  units  of  its  oivn  order,  and  carry  the  quo* 
tient  to  the  next  product. 

Note. — In  multiplj-ing  United  States  money,  care  must  be  taken 
to  point  off  as  many  places  for  cents  and  mills  as  there  are  in  the 
multiplicand. 

1-  Multiply  14  dollars  IG  cents  and  8  mills,  by  5,  6,  and  7. 

$14,168  $14,168  $14,168 

5  6  7 

(2.)  (3.)  (4.) 

$870,40  $894,120  $2141,096 

9  14=i7  X  2  36  =  6  X  6. 


PROOFS    OF   MULTIPLICATION. 

55.  There  are  three  methods  of  proof  for  multiplication  : 

I.  Write  the  multiplier  in  the  place  of  the  multiplicand,  and 
h'.d  the  product  as  before :  if  the  two  products  are  the  same, 
t>6  ycvk  ia  supposed  to  be  right  (Art.  50). 

II,  7>:vide  the  product  by  one  of  the  factors,  and  the  quotient 
"wijl  be  ths^-  ol\\2x  factor. 

HI.  L'y  iSa  method  of  casting  out  the  9's. 

FIRST    METHOD. 

MuLiply  •    -    80432  506 

by   -    -     506         80432 
482592       4048 
402160  2024 

40o98592         1518 

1012 
40698592 

Note  1. — Althougii  wc  generally  begin  the  multiplication  by  tho 
figure  of  the  lowcsi  unit,  yet  we  may  multiply  in  any  order,  if  we 
only  preserve  the  places  of  the  different  orders  of  units.  In  the  ex- 
ample at  the  right,  we  began  with  the  order  of  tens  of  tlicusands  (r 
5tl:  order. 

55.  How  many  proofs  are  there  for  multiplication'!  What  is  the  first  1 
What  is  the  second  1     Wliat  is  the  third  1 


OPERATION. 

641  =  639  + 

2 

232  =  225  + 

7 

4473  +  14 

450 

3195 

1278 

1278 

148698  +  14 

MULTIPLICATION.  59 

2.  Although  either  factor  may  be  used  as  tlie  multiplier  (Art.  50), 
Etill  it  is  best  to  use  that  one  which  contains  the  fewest  figures. 
For,  if  \vc  change  the  process  and  use  the  multiplicand  as  the  mul- 
tiplier,  there  -vvill  be  more  multiplications,  as  shown  in  the  last 
example. 

PROOF    BY    THE    O'S. 

56.  Let  it  be  required  to  multiply  any  two  numbers  together, 
as  641  and  232. 

Analysis. — We  first  find  the  excess  over 
exact  9's  in  both  factors,  and  then  separate 
each  factor  into  two  parts,  one  of  which 
shall  contain  exact  9's,  and  the  other  the 
excess,  and  unite  the  two  by  the  sign  plus. 
It  is  now  required  to  take  639  +  2  =  641, 
as  many  times  as  there  are  units  in 
225  +  7  =  232. 

Every  partial  product,  in  this  multiplica- 
tion,  contains   exact  9's,  except   14,  which 

contains  one  9  and  5  over;  and  as  the  same  may  be  shown  for  any 
two  numbers,  we  see  that, 

If  we  find  the  excess  of  9's  in  each  of  two  factors,  and  then 
multi'ply  them  together,  the  excess  of  9's  in  their  product  will  be 
equal  to  the  excess  of  9's  in  the  product  of  the  factors. 

(1.)  Ex.  (2.)  Ex. 

Multiply         87G03     -    -     6  818327     -    -     2 

by  98G5 ^  9874 1 

Prod.   864203595  -  -  6  80S01G0798  -  -  2 

3.  By  multiplication  we  have 

Ex.4.       Ex.8.     Ex.4.     Ex.  of  product,  2. 
7285  X  143  X  976  =  1016752880. 

Note. — Is  it  necessary  to  commence  the  multiplication  with  the  lowest 
unit  1     Which  factor  is  it  most  convenient  to  use  as  a  multiplier  1 

56.  How  do  you  find  the  excess  of  9's  in  the  product  of  two  factors'! 
If  the  excess  of  9's  in  any  factor  is  0,  what  is  the  excess  of  9's  in  the 
product  1 


60 


MULTIPLICATION. 


Ex.5.     Ex.4.    Ex.0.         Ex.0. 
4.  We  also  have     869  x  49  X  36  =  1532916. 

Note. — When  the  excess  of  9's  in  any  factor  is  0,  the  excess  of  9's 
ill  the  product  is  always  0. 

EXAIIPLES. 

(1.)        (2.)         (3.)        (4.) 
847046     9807602     570409     216987 
8  7         6         9 


(5.) 

(6.) 

(7.) 

103672 

8163021 

90031746 

42   - 

126 

274 

(8.) 

(9.) 

(10.) 

$14,168 

$894,126 

$20034,645 

5 

14 

48 

(11.) 

(12.) 

(13.) 

47321809 

1237506 

437024 

4261 

3460 

400 

(14.) 

(15.) 

(16.) 

8703600 

107030 

30671200 

34600 

5700 

482 

(17.) 

(18.) 

(19.) 

£  s.    d. 

T.  qr.     lb.     oz. 

yd.  ft.    in. 

20  6  8 

3  3  21  14 

16  2  9 

4 

8 

7 

(20.) 

(21.) 

(22.) 

hlid.  gal.    qt.  pt. 

E.F.  qrs.  na. 

12°  42'  55" 

4  42  2  1 

24  2  3 

9 

12 

24 

23.  Mulfiply  IStons  2qrs.  IGlbs.  Ooz.  by  48. 

24.  JMultiply  Oyr.  8ino.  2ivL  Ma.  42m.   by  56. 

25.  Multiply  68   by  the  factors   9   and  8  of  the  composit  > 
number  72. 


MULTIPLICATION.  61 

26.  Multiply  3657  by  the  factors  of  64. 

27.  Multiply  37046  by  the  factors  of  121. 

28.  Multiply  2187406  by  the  factors  of  144 

29.  Multiply  430714934  by  743. 

30.  Multiply  37157437  by  14972. 

31.  Muhiply  47157149  by  37049. 

32.  Multiply  57104937  by  40709. 

33.  Multiply  79861207  by  890416. 

34.  Multiply  9084076  by  9908807. 

35.  Multiply  2748  by  200. 

36.  Multiply  67046  by  10  :  also  by  100. 

37.  Multiply  57049  by  100  :  also  by  1000. 

38.  Multiply  4980496  by  1000  :  also  by  10000. 

39.  Multiply  90720400  by  100  :  also  by  10000. 

40.  Multiply  74040900  by  1  :  also  by  10. 

41.  Multiply  674936  by  100  :  also  by  100000. 

42.  Multiply  478400  by  270400. 

43.  Multiply  367000  by  37409000. 

44.  Multiply  7849000  by  84694000. 

45.  Multiply  89999000  by  97770400. 

46.  Muhiply  9187416300  by  274987650000. 

47.  Multiply  86543291213456  by  12637482965. 

48.  Multiply  76729835645873  by  217834569. 

49.  If  it  costs  2479  dollars  to  build  one  mile  of  plank  road, 
bow  much  would  it  cost  to  build  25  miles  ? 

50.  How  far  would  a  vessel  sail  in  9  days,  of  24  hours  each, 
at  the  rate  of  15  miles  an  hour  ? 

51.  A  man  bought  two  farms,  one  of  125  acres,  at  26  dollars 
an  acre  ;  another  of  96  acres,  at  32  dollars  an  acre  ;  he  paid  at 
one  time  2500  dollars ;  at  another  time  1725  dollars  :  what 
remained  to  be  paid  ? 

52.  In  9  pieces  of  kersey,  each  containing  lAyd.  3qr.  2na., 
how  many  yards  ?* 


*  NorE. — When  the  multiplicand  is  a  compound  denominate  number, 
and  the  multiplier  a  composite  number,  it  is  best  always  to  multiply  by  the 
factois  of  the  composite  number. 


62  MULTIPLICATION. 

53.  Wliat  will  15  gallons  of  wine  cost  at  5s.  S^d.  per  gallon  ? 

54.  What  will  be  the  value  of  416  sheep  at  $2,48  a  head  ? 

55.  Bought  40  barrels  of  flour  at  $8,75  a  barrel,  and  sold 
them  for  $9,121  a  barrel :  what  was  the  whole  gain  ? 

56.  What  is  the  weight  of  11  hogsheads  of  sugar,  each 
weighing  7cwL  2qr.  18lb.,  and  what  would  be  its  value  at  6 
cents  a  pound  ? 

57.  A  merchant  bought  36  pieces  of  broadcloth,  each  piece 
containing  44  yards,  at  4  dollars  a  yard :  what  did  the  whole 
cost? 

58.  A  gentleman  whose  annual  income  is  $3479,  expends  for 
pleasure  and  travelling  $600  ;  for  books  and  clothing  $570  ;  for 
board  and  other  expenses  $1200  :  how  much  will  he  save  in 
5  years  ? 

59.  The  number  of  milch  cows  in  the  state  of  New  York  in 
1850  was  931324 :  Avhat  would  be  their  value  at  18  dollars 
each  ? 

60.  If  a  man  travel  20/?u*.  ofur.  IQrd.  in  one  day,  how  far 
will  he  travel  in  24  days  ? 

61.  How  long  will  it  take  a  man  to  mow  14  acres  of  grass, 
allowing  10  working  hours  a  day,  if  he  mow  one  acre  in  4Jir. 
4omt.  SOsec.  ? 

62.  If  a  man  spend  six  cents  a  day  for  segars,  how  mucli 
will  he  spend  in  thirty  years,  allowing  three  hundred  and  sixty- 
five  days  to  the  year  ? 

63.  A  farmer  sold  118  bushels  of  barley  for  62^  cents  a 
bushel,  and  receives  5  barrels  of  flour  at  $9,87^  a  barrel,  and 
the  remainder  in  cash  :  how  much  cash  did  he  receive  ? 

64.  Two  persons  start  at  the  same  point  and  travel  in  oppo- 
site directions,  one  at  the  rate  of  34  miles  a  day,  the  other  at 
the  rate  of  28  miles  a  day :  how  far  apart  will  they  be  at  the 
end  of  14  days? 

65.  An  apothecary  sold  8  bottles  of  laudanum,  each  con- 
taining 10  T  6  5  2  9  14^7*. :  what  was  the  weight  of  the 
whole  ? 

GO.  A  farmer  took  7  loads  of  oats   to  market,  each   load 


MULTIPLICATION.  63 

having  20  bags,  and  each  bag  containing  2hush.  opk.  Qqi. :  how 
many  bushels  of  oats  did  he  take  to  market  ? 

67.  If  in  a  woollen  factory  4G8  yards,  of  cloth  are  made  in 
one  day,  how  many  yards  will  be  made  in  313  days  ? 

G8.  The  greatest  number  of  whales  ever  captured  in  the 
northern  seas,  in  one  season,  was  2018.  Estimating  the  oil 
produced  from  each  to  have  been  212  barrels,  what  was  the 
amount  of  oil  produced  ? 

69.  What  will  be  the  value  of  an  ox  weighing  7cwf.  2qr. 
IQlh.,  at  11  cents  a  pound  ? 

70.  What  will  be  the  cost  of  245  hogsheads  of  sugar,  each 
weighing  984  pounds,  at  7  cents  a  pound  ? 

71.  Bought  6  loads  of  hay,  each  weighing  18cw^  Sqis.  21/6. ; 
after  letting  a  neighbor  have  2tons  Ibcwt.  Iqr.  bib,,  how  much 
will  there  be  Left  ? 

72.  In  an  orchard  there  are  136  apple  trees,  each  tree  yield- 
ing 17  bushels  of  apples :  how  many  bushels  did  the  whole 
orchard  yield,  and  what  would  they  be  worth  at  42  cents  a  bushel? 

73.  A  flour  mei'chant  bought  1845  barrels  of  flour  at  7  dol- 
lars per  barrel.  He  sold  at  one  time  528  barrels,  at  9  dollars  a 
barrel ;  at  another  time  856  barrels  at  8  dollars  a  barrel ;  how 
many  barrels  had  he  left,  and  at  what  cost  ? 

74.  What  are  25  hogsheads  of  sugar  worth,  each  weighing 
872  pounds,  at  6^  cents  a  pound  ? 

75.  It  is  estimated  that  the  whole  amount  of  land  appropriat- 
ed by  the  General  Government  for  educational  purposes,  to 
the  1st  of  January,  1854,  was  52770231  acres.  What  would 
be  the  value  of  this  land  at  the  Government  price  of  one  dollar 
and  twenty-five  cents  an  acre  ? 

76.  If  30  men  can  do  a  piece  of  work  in  25  days,  how  long 
will  it  take  one  man  to  do  it  ? 

77.  A  man  desired  that  his  property  should  be  equally  divided 
omono-  his  5  children,  giving  each  twenty-seven  hundred  dol- 
lars :  what  was  the  amount  of  his  property  ? 

78.  Bought  9  chests  of  tea,  each  containing  72  pounds,  at 
37:^  cents  a  pound :  wha/  was  the  cost  of  the  whole  ? 


64:  MULTIPLICATION. 

79.  A  farm  consisting  of  127  acres,  was  sold  at  auction  foi 
S37,565  an  acre  :  what  sum  of  money  did  it  bring  ? 

80.  A  drover  bought  127  head  of  beef  cattle  at  an  average 
of  39  dollars  per  head ;  he  sold  86  of  them  for  43  dollars  per 
head;  for  how  much  per  head  must  he  sell  the  remainder,  to 
clear  on  the  first  cost  1246  dollai's  ? 

81.  What  will  75  firkins  of  butter  cost,  each  firkin  weighing 
56  pounds,  at  16  cents  a  pound? 

82.  A  merchant  bought  a  box  of  goods  containing  37  pieces, 
each  piece  containing  46  yards,  worth  7  dollars  a  yard :  what 
did  the  box  of  goods  cost  ? 

..  83.  A  bond  was  given  April  20th,  1850,  and  was  paid  Sept. 
4th,  1856  :  what  will  be  the  product,  if  the  time  which  elapsed 
from  the  date  of  the  bond  to  the  time  it  was  paid  be  multiplied 
by  45  ? 

84.  "What  distance  will  a  Wheel  16  feet  8  inches  in  circum- 
ference measure  on  the  ground,  if  rolled  over  84  times  ? 

85.  What  is  the  difference  between  twice  eight  and  fifty,  and 
twice  fifty-eight  ? 

86.  How  much  wood  in  4  piles,  each  containing  5  cords, 
6  cord  feet  and  32  cubic  feet  ? 

87.  A  man  bought  56  acres  of  land  for  $25  an  acre,  and  94 
acres  for  $32  an  acre ;  if  he  sells  the  whole  at  $30  an  acre, 
will  he  gain  or  lose,  and  how  much  ? 

88.  If  12  men  can  build  a  wall  in  16  days,  how  many  men 
will  be  required  to  build  a  wall  nine  times  as  long  m  half  the 
time  ? 

89.  A  former  sold  4  cows  for  $25,50  each;  12  sheep  for 
$2,12-1^  each ;  and  3  calves  for  $7,25  each ;  what  was  the 
amount  of  the  sale  ? 

90.  If  it  requires  116  tons  of  iron  to  construct  one  mile  of 
railroad,  how  much  would  it  require  to  construct  a  railroad 
from  Albany  to  Buff;ilo,  it  being  326  miles? 

91.  A  merchant  bought  960  pounds  of  cheese  at  9  cents  a 
pound  ;  148  pounds  of  butter  at  12^  cents  a  pound.  He  gave 
in  payment,  12  yards  of  cloth,  at  $4,75  a  yard  ;  186  pounds  of 


MTTLTIPLICATION, 


65 


sugar  at  7  cents  a  pound,  and  the  remainder  in  cash  :  how  much 
cash  did  he  pay  ? 

92.  If  a  family  consume  12^'/.  25-1;.  \pt.  of  ale  in  a  week, 
how  much  will  they  consume  in  14  weeks? 

93.  How  much  brandy  will  supply  an  army  of  25,000  men 
for  one  month,  if  each  man  requires  Igal.  2qf.  Ijyt.  2gi. 

94.  It  is  estimated  that  the  French,  during  the  years  1854 
and  1855,  transported  to  the  Crimea  80000  horses,  and  that 
70000  of  them  were  lost  in  the  same  time.  Supposing  the 
first  cost  of  each  horse  to  be  $100,  and  the  cost  of  transporta- 
tion $95  per  head,  what  was  the  value  of  the  horses  lost  ? 

95.  A  man  purchased  a  piece  of  woodland  containing  27 
acres,  at  39  dollars  per  acre ;  each  acre  produced  on  an  average 
70  corda  of  wood,  which,  being  sold,  yielded  a  nett  profit  of  45 
cents  a  cord  :  how  much  did  the  profit  on  the  wood  fall  short  of 
paying  for  the  land  ? 

BILLS    OF    PAKCELS. 

9G.  Chicago,  June  10,  1856. 

J/r.  John  C.  Smith,  Bought  of  David  Toombs. 

14  pounds  of  tea,  at  75  cents,  -         -         $ 


9       " 

"  coffee. 

14     " 

42       « 

"  sugar, 

11     " 

3       " 

"  pepper, 

12^  " 

5       " 

"  chocolate. 

56    " 

12       " 

"  candles. 

16    " 

Received  payment, 

David  Toombs. 

97  New  York,  March  20th,  1857. 

Mr.  Jacob  Johns,                 Bought  of  George  Bliss  Sf  Co 
48  pounds  of  sugar  at  9i  cents  a  pound,     -     -     -     $ 
6  hogs,  of  molasses,  each  containing  63  gallons, 
at  27  cents  a  gallon, 

8  casks  of  rice,  285  lbs.  each,  at  5  cts.  a  pound, 

9  chests  of  tea,  86  lbs.  each,  at  87^  cts.  a  pound, 

4  bags  of  coffee,  each  67  lbs.,  at  11  cts.  a  poiuid, 


Received  payment,  ^ 

4  Geo.  Bliss  &'.  C-o- 


66  MULTIPLICATION. 

98.  Hartford,  Novembet  21s<,  1856. 

Gideon  Jones,  Bought  of  Jacoh  Thrifty. 

78  chests  of  tea,  at  $55,G5  per  chest,       -         -         % 
251  bags  of  coffee,  100  pounds  each,  at) 
12^  cts.  per  pound,     -         -  ) 

317  boxes  of  raisins,  at  $2,75  per  box, 
1049  barrels  of  shad,  at  $7,50  per  barrel,   -         -         - 
76  barrels  of  oil,  32  gallons  each,  at  $1,08  per  gal., 


Amount,  $ 
Received  the  above  in  full,  Jacob  Thrifty. 

99.  Baltimore,  Jan.  \st,  1855. 
Mr.  Abel  Wirt,  Bought  of  Timothy  Stout. 

10  yards  of  broadcloth,  at  $4,37i,  -         -         -         $ 
75       "     "    sheeting,      "       ,09     - 
42       "     "    plaid  prints  ''       ,45     - 

5  barrels  of  Genesee  flour,  at  $7,87^,    - 

7  pairs  of  boots,  at  $1,60  per  pair, 
18  bushels  of  corn,  at  72  cts.  per  bushel, 

"$ 

100.  Montreal,  Oct.  16th,  1855. 
3fr.  Chas.  Snow,  Bought  of  Vose,  Duncan  Sf  Co. 

45  yai-ds  of  broadcloth  at    9s.  M.    -  -  £  s.  d. 

56      "  "  "    12s.  ^d.  - 

16      "          vesting?,     "      6s.  8^d.  -  -  - 

24  lbs.  colored  thread,  "      os.  Ad.    -  -  - 

72  pairs  silk  hose,         "      7s.  5|(/.  - 

108  yards  carpeting,        "    14s.  10c?.  -  -  - 

Received  payment,  £ 

VosE,  Duncan  &  Co. 


DIVISION. 


67 


DIVISION 

57.  Division  is  the  operation  of  finding  from  two  munhers  a 
ihird,  which  midtiplied  by  the  first,  to  ill  produce  the  second. 

The  first  number,  or  number  by  which  we  divide,  is  called 
the  divisor. 

The  second  number,  or  number  to  be  divided,  is  called  the 
divide7id. 

The  third  number,  or  result,  is  called  the  quotient. 

The  quotient  shows  how  many  times  the  dividend  contains 
the  divisor. 

When  the  quotient  is  expressed  by  an  integral  number,  the 
division  is  said  to  be  exact.  When  it  cannot  be  so  expressed, 
the  part  of  the  dividend  that  is  undivided,  is  called  the  remainder. 

58.  There  are  always  three  numbers  in  every  division,  and 
sometimes  four  :  1st.  the  dividend ;  2d.  the  divisor  ;  od.  the 
quotient ;  and  4th,  the  remainder. 

There  are  three  methods  of  denoting  division  ;  they  are  the 
following  : 

12  -r  3  expresses  that  12  is  to  be  divided  by  3. 
-L2  expresses  that  12  is  to  be  divided  by  3. 

3)12        expresses  that  12  is  to  be  divided  by  3. 
When  the  last  method  is  used,  if  the  divisor  does  not  exceed 
12,  we   draw  a  line  beneath  the  dividend   and  set  the  quotient 
under  it.     If  the  divisor  exceeds  12,  we  draw  a  curved  line  on 
the  right  of  the  dividend,  and  set  the  quotient  at  the  right. 

59.  Short  Division  is  the  operation  of  dividing  when  the 
work  is  performed  mentally,  and  the  results  only  written  down. 
It  is  limited  to  the  cases  in  which  the  divisors  do  not  exceed  12. 


57.  What  is  division  l  What  is  the  number  to  be  divided  called  1  What 
is  the  number  called  by  which  we  divide  1  What  is  the  answer  called  1 
What  is  the  number  called  which  is  left  1 

58.  How  many  parts  are  there  in  division  1  Name  them.  How  many 
eigns  arc  there  in  division  1     Mako  and  name  them. 

59.  What  is  short  division  ?  How  is  it  generally  performed  1  Where  is 
the  quotient  written  ]     To  what  cases  is  it  limited  1 


68  DIYISTON. 

1.  Divide  45G  by  4. 

Analysis. — Tlie  number  456  is  made   up   of  4  hundreds,  5  tens, 
?nd  6  units,  each  of  which  is  to  be  divided  by  4. 
Dividing  4  hundreds  by  4,  we  have  the  quotient,         operation. 
1  hundred  :  5  tens  divided  by  4,  gives  1  ten  and  4)456 

1  ten  over  :  reducing  this  to  units  and  adding  in  114 

die    6,    we   have    16    units,    which  contains  4, 
4  times:  hence,  the  quotient  is  114:  that  is,   the  dividend  contains 
the  divisor  114  times. 

2.  Divide  £11  8s.  Id.  Zfar.  by  5. 

Analysis. — Dividing  £11   by  5,   the  quofient  is  £2  and  £1  re- 
maining.    Reducing  this  to  shillings  and  adding 
in  the  8,  we  have  28s.,  which  divided  by  5,  gives         operation. 
5s.  and  35.  over.     This    being  reduced  to  pence         £     s.     d.  far. 
and   Id.   added,  gives   43J.     Dividing  by  5,  we     5)11     8      7     3 
have  ^d.  and  3c?.  remainder.      Pteducing  2d.  io  2      5     8     3 

farthings,  adding  3  farthings,  and  again  dividing 
by  5,  gives  the  last  quotient  figure  3/ar. 


3.  Divide  £6  85.  M.  by  8.  operation, 

£     s      d 
Here  we  have  to  pass  to  shillings  before 

making  the  first  division. 


S)6     8     8 


16     1 


4.  Divide  11772  by  327. 


Analysis. — Flaving  set  down  the  divisor  on  the  left  of  the  dividend, 
it  is  seen  that  327  is  not  contained  in  the  first 
three  iigvues  on  the  left,  which  are  117  hundreds.  opekation. 

But    by    observing   that    3  is   contained   in  11,        327)11772(36 
3  times  and  something  over,  we  conclude  that  the  981 

divisor  is  contained  af  least  3  times  in  the  first  1962 

four  figures,  1177  tens,  which  is  a  pnrtialdividcnd.  1962 

Set  down  the  quotient  figure  3,  and  multiply  the 
divisor  by  it :  we  thus  get  981  tens,  which  being  less  than  1 177.  the 
quotient  figure  is  not  too  great:  we  subtract  the  981  tens  from  the 
first  four  figures  of  the  dividend,  and  find  a  remainder  196  tens, 
which  being  less  than  the  divisor,  the  quotient  figure  is  not  too  small. 
Reduce  this  remainder  to  units  and  add  in  the  2,  and  we  have  1962. 

As  3  is  contained  in  19,  6  times,  we  conclude  that   the  divisor  is 
contained  in  1962  as   many  as  6  times.     Setting  down  6  in  the  quo- 


DIVISION.  69 

tient  and  mnltipJymg  the  divisor  by  it,  we  find  the  product  to  be 
19C2.  Therefore  the  entire  quotient  is  36.  or  the  divisor  is  contained 
36  times  in  the  dividend. 

60.  From  the  above  analysis,  we  have  the  following  rule  for 
the  division  of  numbers. 

1.  Begin  icith  the  highest  order  of  miits  of  the  dividend,  and 
pass  on  to  the  lower  orders  until  the  fewest  nmnher  of  figures  he 
found  that  will  contain  the  divisor :  divide  these  figures  by  it  for 
the  first  figure  of  the  quotient :  the  unit  of  this  figure  will  be  the 
same  as  that  of  the  lowest  order  in  the  partial  dividend. 

II.  Mnhiplg  the  divisor  by  the  quotient  figure  so  found,  and 
subtract  the  product  from  the  partial  dividend. 

III.  Reduce  the  remainder  to  units  of  the  next  loiver  order, 
and  add  in  the  units  of  tJiat  order  found  in  the  dividend:  this 
gives  a  new  partial  dividend.  Proceed  in  a  similar  manner  until 
units  of  every  order  shdl  have  been  divided. 

DIRECTIONS    FOR    THE    OPERATIONS. 

Notes. — There  are  tivc  operations  in  Long  Division.  1st.  To  write 
down  the  numbers  :  2d.  To  divide,  or  find  how  many  times  :  3d.  Tc 
multiply:  4th.  To  subtract:  5th.  To  bring  down,  to  Ibrm  the  partia 
dividends. 

2.  The  product  of  a  quotient  figure  by  the  divisor  must  never  b 
larger  than  the  corresponding  partial  dividend  :  if  it  is,  the  quotien 
figure  is  too  large  and  must  be  diminished. 

3.  When  any  one  of  the  remainder.s  is  greater  than  the  divisor,  the 
quotient  figure  is  too  Fmall  and  must  be  increased. 

4.  Tlie  unit  of  any  quotient  figure  is  the  same  as  that  of  the  partial 
dividend  from  which  it  is  obtained.  The  pupil  should  always  name 
the  unit  of  every  quotient  figure. 

60.   Give  the  rule  for  the  division  of  numbers. 

Notes. — 1.  How  many  operations  are  there  in  division  1     Name  them. 

2.  If  a  partial  product  is  greater  than  the  partial  dividend,  what  does  it 
indicate  '^     What  then  do  you  do  ? 

3.  What  do  you  do  when  an}'  one  of  the  remainders  is  greater  than  the 
divisor  '\ 

4.  What  is  the  unit  of  any  figure  of  the  quotient  \  \^'hen  the  divisoi 
18  contained  in  simple  units,  what  will  be  the  unit  of  tlic  quotient  figure 


70  DIVISION. 

0.  If  any  partial  dividend  is  less  than  the  divisor,  the  correspond 
ing  quotient  figure  is  0. 

6.  When  there  is  a  remainder,  after  division,  write  it  at  the  right 
of  tiie  quotient,  and  place  the  divisor  under  it. 

PRINCIPLES    RESULTING    FROM    DIVISION. 

1.  When  the  divisor  is  equal  to  the  dividend,  the  quotient  will  he  1. 

2.  When  the  divisor  is  less  than  the  dividend,  the  quotient  will  ha 
greater  than  1.     The  quotient  will  be  as  many  times  greater  than  1 
as  the  dividend  is  times  greater  than  the  divisor. 

3.  Wlien  the  divisor  is  greater  than  the  dividend,  the  quotient  will 
be  less  than  1.  The  quotient  will  be  such  a  part  of  1,  as  the  divi- 
dend is  of  the  divisor. 

4.  When  the  divisor  is  1,  the  quotient  will  be  equal  to  the  divi- 
dend. 

PROOF. 

61.  There  are  two  methods  of  proof  for  division :  1st.  By 
multiplication ;  2d.  By  the  excess  of  9's. 

FIRST    METHOD. 

By  the  definition  of  division,  the  quotient  is  such  a  number 
as  multiplied  by  the  divisor  will  produce  the  dividend  (Ai-t.  57). 

In  example  4,  each  product  of  the  divisor  by  a  figure  of  the  quo- 
tient is  a  partial  product,  and  the  sum  of  these  products  is  the  product 
of  the  divisor  and  quotient  (page  57,  Note).     Each  product  is  taken, 

M'hcn  it  is  contained  in  tens,  what  will  be  the  unit  of  the  quotient  figure  1 
Wlien  it  is  contained  in  hundreds'!     In  thousands] 

5.  If  any  partial  dividend  is  less  than  the  divisor  what  is  the  correspond- 
ing figure  of  the  quotient  ? 

6.  When  there  is  a  remainder  after  division,  what  do  you  do  witli  it? 
Note. — 1.  When  the  divisor  is  equal  to  the  dividend,  what  will  the  quo- 
tient be  1 

2.  When  the  divisor  is  less  than  the  dividend,  how  will  the  quotient 
compare  with  1  1     How  many  times  will  it  be  greater  than  1  ? 

3.  When  tlie  divisor  is  greater  than  the  dividend,  how  will  the  quotient 
compare  with  1  ?    What  part  will  the  quotient  be  of  1  ? 

4.  When  the  divisor  is  1,  what  will  the  dividend  be  ! 

61.  How  many  methods  of  proof  are  there  for  division  !  What  are  they  ! 
What  is  the  proof  by  multiplication  !     What  is  the  proof  by  the  9's  ? 


DIVISION.  71 

geparately,    from    the  dividend,    and    nothing  operation 

remains.     But,  taking  each  product  away,  in  327)11772(36 

succession,  leaves  the  same  remainder  as  would  1981 

be  left  if  their  sum  were  taken  away  at  once.  1962 

Hence,   the   number   36,  when   multiplied   by  1962 
the  divisor  327,  gives  a  product  equal  to  the 

dividend    11772;  therefore,  36   is  the  quotient  (Art.  57):  hence,  to 
prove  division, 

Blidtiply  tlie  divisor  hy  the  quotient  and  add  in  the  remainder, 
if  any.  If  the  work  is  right,  the  result  will  be  the  same  as  the 
dividend. 

Note. — Divide  325  by  19.  The  quotient  is  17,  and  16  remainder: 
the  true  quotient  is  17^|  ,  for.  this  being  multiplied  by  the  divisor  19, 
will  give  the  dividend.  If  the  pupil  knew  how  to  dispose  of  the  frac- 
tional part,  we  should  simply  say,  "  Multiply  the  divisor  by  the 
quoticntj^'  which  is  exactly  what  we  do  under  the  rule. 

PROOF    BY    9's. 

Since  the  dividend  is  equal  to  the  product  of  the  divisor  and 
quotient,  it  follows  that  if  the  excess  of  9's  in  the  divisor  be 
multiplied  by  the  excess  of  the  9's  in  the  quotient,  the  excess 
of  9's  in  the  product  will  be  equal  to  the  excess  of  9's  in  the 
dividend  (Art.  56).     Hence, 

Find  the  excess  of  9's  in  the  divisor  and  in  the  quotient :  mul- 
tiply them  together,  and  note  the  excess  of  9's  in  the  product :  if 
this  is  equal  to  the  excess  of  9's  in  the  dividend,  the  work  may  he 
regarded  as  Hght. 

Divisor,  327,  excess  of  9's  -         -      3  ] 

Quotient,  36,      «        -         -         .      o\  ^''^^^^^>  ^' 

Dividend,  11772,       "         -         -         -       0 

EXAMPLES. 

(1.)        (2.)         (3.)  (4.) 

3)19737     4)147308     5)1346840     6)1650930 

(5.)  (6.)  (7.) 

6)47689872      9)10324683      7)506321494 


72 


DIVISION. 


£ 

3)47 


(8.) 

s. 
19 


A. 
9)37 


(9.) 
i?. 
3 


P. 

17 


(11.) 

$        cts. 
8)634     75 

6 

(10.) 

yd.  qr.  na. 
5)47     3     1 


(12.) 

$       cts.    m. 
7)1468     0     96 


(13.) 

$ 
12)802346 


16 


Divide  $29,25  by  26. 
Divide  $10,125  bv  27. 
Divide  $347,49  by  429 
Divide  $751,50  by  150. 
Divide  $571 1,04  by  108. 
Divide  $315  by  $35. 
Divide  $50065  by  $527. 
Divide  $432  by  54. 


14.  Divide  734947644  by  48.         22. 

15.  Divide  8536752  by  36.  23. 

16.  Divide  3367598  by  19.  24. 

17.  Divide  49300  by  725.  25. 

18.  Divide  6477150  by  145.         26. 

19.  Divide  770  by  28.  27. 

20.  Divide  $87,256  by  5.     28. 

21.  Divide  $495,704  by  129.    29. 

30.  Divide  334422198  by  438. 

31.  Divide  714394756  by  1754. 

32.  Divide  47159407184  by  3574. 

33.  Divide  5719487194715  by  45705. 

34.  Divide  4715714937149387  by  17493. 

35.  Divide  671493471549375  by  47143. 

36.  Divide  571943007145  by  37149. 

37.  Divide  1714347149347  by  57143. 

38.  Divide  49371547149375  by  374567. 

39.  Divide  171493715947143  by  571007. 

40.  Divide  6754371495671594  by  678957. 

41.  Divide  7149371478  by  121. 

42.  Divide  71900715708  by  57149. 

43.  Divide  14714937148475  by  123456. 

44.  Divide  729^.  2R.  IP.  by  41. 

45.  Divide  oQMa.  Qhr.  by  240. 

46.  Divide  1298mi.  2fur.  33rd.  by  37. 

47.  Divide  95Md.  Ggal  by  120. 

48.  Divide  232bush.  3ph.  Igt.  by  105. 

49.  Divide  $18306,25  by  725. 

50.  Bonp;ht  7  yds.  of  clDtli  for  1  6.«.  Ad. :  wlint  did  it  cost  per  yd.  ? 


DIVISION.  73 

51.  A  man  travelled  2Gomi.  Gfur.  IQrd.  in  12  clays  :  how  far 
did  he  travel  in  one  day? 

52.  If  569^.  2Ii.  23P.  be  equally  divided  between  9  per- 
sons, how  much  will  5  of  them  have  ? 

53.  The  annual  income  of  a  gentleman  is  $10000:  how 
much  is  that  per  day,  counting  365  days  to  the  year  ? 

54.  What  number  multiplied  by  9999  will  produce  987551235  ? 

55.  A  gentleman  owning  an  estate  of  $75000,  gave  one-fourtL 
of  it  to  his  wife,  and  the  remainder  was  divided  equally  among 
his  five  children  :  liow  much  did  each  receive  ? 

56.  The  expenditure  of  the  United  States  for  1853  was 
$54026818  :  how  much  Avould  that  be  per  day,  allowing  365 
days  to^the  year  ? 

57.  If  28  yai'ds  of  cloth  cost  $133,  what  will  one  yard  cost? 

58.  If  I  pay  $637,50  for  51  yards  of  cloth,  what  is  the  price 
per  yard  ? 

59.  The  city  of  New  York,  in  1850,  had  104  periodical  i)ub- 
lications,  with  an  aggregate  circulation  of  78747600  copies: 
what  would  be  the  average  circulation  of  each  ? 

60.  Bought  19. bushels  of  wheat  for  $30,875  :  what  was  the 
cost  of  one  bushel  ? 

61.  How  long  will  9125  loaves  of  bread  last  5  families,  if 
each  family  consume  5  loaves  a  day  ? 

62.  The  product  of  two  numbers  is  7207272072,  and  the 
multiplier  9009  :  what  is  the  multiplicand  ? 

63.  How  many  rings,  each  weighing  Adwt.  12g7\,  can  be 
made  from  lOor.  11  dwt.  12gr.  of  gold  ? 

64.  If  iron  is  worth  2  cents  a  pound,  how  much  can  be 
bought  for  $67,50  ? 

65.  If  14  sticks  of  hewn  timber  measure  12  2^.  38/;!.  118m., 
how  much  does  each  stick  contain  ? 

66.  In  1850,  Pennsylvania  manufactured  285702  tons  of  pig 
iron,  and  employed  9285  hands  :  what  was  the  average  product 
of  each  hand  ? 

67.  The  number  of  collesre  libraries  in  the  United  States  in 
1850,  was  213,  containing  942321  volumes :  what  would  be  the 

average  number  of  volumes  in  each  ? 

■1 


74  OONTEAOTIONS 


CONTRACTIONS  AND  APPLICATIONS. 

CONTRACTIONS   IN   MULTIPLICATION. 

62.   Contractions  in  Multiplication  are  short  methods  of  find- 
ing products  when  the  multipliers  are  particular  numbex'S. 

63.    To  multiphj  hy  25. 

1.  Multiply  356  by  25. 

Analysis. — If  we  annex  two  ciphers  to  the  niul-       operation. 
tiplicand,   we  multiply  it   by  100   (Art.  5a):  this         4)35600 
product  is  4  times  too  great;  for  the  multiplier  is  8900 

but  one-fourth  of  100  ;  hence,  to  multiply  by  25, 

Annex  two  ciphers  to  the  multiplicand  and  divide   the  result 
hy  4:. 

EXAMPLES. 


2.  IMultiply  287  by  25. 

3.  Multiply  184  by  25. 


4.  IMultiply  6741  by  25. 

5.  Multiply  3074  by  25. 


OPERATION 

15 

3^ 

3 

45 

48  Ans 

64.    When  the  multiplier  contains  a  fraction. 

AVhat  is  the  jjroduct  of  15  multiplied  by  31  ? 

Analysis. — The  multiplicand  is  to  be  taken 
3  and  one-fifth  times  :  taking  it  one-fifth  times, 
gives  3,  wliich  we  write  in  the  units  place  : 
then,  taking  it  3  times,  gives  45,  and  the  sum 
48  IS  the  product ;  hence 

Rule. — Take  such  a  pari  of  the  multi- 
plicand as  the  fraction  is  of  1 ;  then  multiply  hy  the  integral 
number,  and  the  sum  of  the  products  will  be  the  required 
product. 

2.  Multiply  327  by  8.}.  5.  Multiply  1272  by  12|. 

3.  Multiply  23474  by  16^.      I      6.  Multiply  9824  by  272^. 

4.  Multiply  34700  by  127"^.     |     7.  Multiply  3828  by  731. 

62.  Wliat  are  contractions  in  multiplication  1 

G3    How  do  you  multiply  by  251 

C4.  How  do  you  multiply  when  the  multiplier  contains  a  fraction! 


IN    MULTIPLICATION. 


76 


65.    To  multiplij  hj  12?,. 
1.  Multiply  286  by  12J. 
Analysis. — Since  121  is  one-eighth  of  100, 
Annex  two  ciphers  to  the  multiplicand  and  divide 
the  result  by  8. 

EXAMPLES. 


OPERATION. 

8)28600 


1.  Multiply  384  by  121. 

2.  Multiply  476  by  12^. 


3. 


Multiply     14800  by  12^ 
4.  Multiply  G70418  by  12-1. 

G6.   To  multiply  hy  331. 
1.  Multiply  975  by  33 J. 

Analysis. — Aunoxiiig  Uxo  cipliers  to  the 
multiplicand,  multiplies  it  by  100:  but  the 
multiplier  is  one-third  of  100  :  hence, 

Annex  two  ciphers  and  divide  the  result  by  3. 


operation. 
3)97500 
32500 


EXAMPLES. 

1.  Multiply  1670252  by  33i 

2.  Multiply  1480724  by  33i 

67.    2"o  multiply  hy  125. 
].  Multiply  1125  by  125. 

Analysis. — Annexing  three  ciphers  to  the 
multiplicand,  multiplies  it  by  1000  :  but  125 
is  but  one-eighth  of  one  thousand  :  hence, 

Annex  three  ciphers  and  divide  the  result  by  8 

EXAMPLES. 


3.  Multiply  10675512  by  331. 

4.  Multiply    4442172  by  33i. 


OPERATION. 

8)1125000 
140625 


1.  Multiply  59264  by  125. 

2.  Multiply  17593408  by  125. 


3.  Multiply  1940812  by  125. 

4.  Multiply     140588  by  125. 


COXTRACTIO^VS  IN   DIVISION. 
68.   Contractions  in   Division  are  short  methods  of  finding 
the  quotient,  when  the  divisors  are  particular  numbers. 

65.  How  do  you  multiply  by  12.|  1 

66.  How  do  you  multiply  by  33^1 

67.  How  do  you  multiply  by  125  ] 

68.  What  are  Contractions  in  Division? 


76  CONTKACTIONS 

69.  By  revei'siiig  the  last  four  processes,  we  have  the  four 
following  rules  : 

1.  To  divide  any  number  by  25  : 

Multiply  the  number  by  4,  and  divide  the  product. by  100. 

2.  To  divide  any  number  by  12i  : 

Multiply  the  number  by  8,  and  divide  the  product  by  100. 

3.  To  divide  any  number  by  33  J  : 

Multiply  the  number  by  3,  and  divide  the  product  by  100. 

4.  To  divide  any  number  by  125  : 
Multiply  by  8,  and  divide  the  product  by  1000. 

EXAMPLES. 


1.  Divide        6350  by  25. 

2.  Divide      21345  by  25. 

3.  Divide    G5G280  by  25. 

4.  Divide  7278675  by  25. 

5.  Divide  5287215  by  25. 

6.  Divide       12225  by  12^. 

7.  Divide      10650  by  12i. 

8.  Divide      11925  by  121. 


9.  Divide  1760000  by    12^. 

10.  Divide  67500  by    331. 

11.  Divide  1308400  by    331. 

12.  Divide  15851400  by    331. 

13.  Divide  8072400  by    33J. 

14.  Divide  281250  by  125. 

15.  Divide  6015750  by  125. 

16.  Divide  2026875  by  125. 


70.    Wlien  the  divisor  is  a  composite  number. 

1.  How  many  feet  and  yards  are  there  in  288  inches  ? 

Analysis. — Since  there  are   12   inches  in  1  foot,  there  will  be  as 
many  feet  in  288  inches  as  12  is  contained  times 
in  288  ;  viz.,  24  feet,  in  which  the  unit  is  1  foot.  operation. 

Since  3  feet  make  1  yard,  there  will  be  as  many  12)288 

yards  in  24  feet  as  3  is  contained  times  in  24  ;  3)24 

viz.,  8  yards  :  in  which  the  unit  is  1  yard.     We  8 

have   thus  passed,  by  division,  from  the  unit  1 
inch  to  the  unit  1  foot,  and  then  to  the  unit  1  yard  ;  that  is,  in  each 

69.  What  rules  do  you  get  by  reversing  the  four  previous  rules  ?  Gito 
them. 

70  What  is  a  composite  number  1  Under  ".<o\v  ninny  points  of  view 
may  division  be  regarded  1  What  arc  they  !  A'liat  is  the  rule  for  division 
when  the  divi.sor  i~  ;>  romposile  nunibor '  When  there  are  remainders 
after  division,  how  do  -ov   And  tli    remainder  in  units  of  the  dividenrl  ■" 


IN    DIVISION.  77 

operation,  we  have  increased  the  unit  as  many  times  as  there  are  units  in 
the  divisor. 

Let  us  now  use  the  same  numbers,  ii  an  entirely  difFerent  question.' 

2.  If  288  dollars  be  equally  divided  among  3G  men,  what 
will  be  the  share  of  each  ? 

Analysis. — Since  288  dollars   is   to     36  =  12  x  3     operation. 
be  equally  divided  among  36  men,  each  12)288 

■will  liavo  as  many  dollars  as  36  is  con-  3)24 

tained  times  in  288.   Dividing  288  into  8 

12  equal  parts,  we  find  that  each  part 

is  24  dollars.  If  each  of  these  parts  be  now  divided  into  3  equal 
parts,  there  will  then  be  36  parts  in  all,  each  equal  to  8  dollars  : 
here,  the  unit  of  the  result  is  the  same  as  that  of  the  dividend.  Hence, 
we  may  regard  division  under  two  points  of  view  : 

1.  As  a  process  of  reduction,  in  which  the  unit  of  each  suc- 
ceeding dividend  is  increased  as  many  times  as  there  arc  units  in 
the  divisor : 

2.  As  a  process  of  separating  a  number  into  equal  parts  ;  in 
%vhich  case  the  unit  of  a  part  will  he  the  same  as  that  of  the 
dividend. 

Hence,  the  follow^ing  rule  when  the  divisox*  is  a  composite 
number : 

RuLii;. — Divide  the  dividend  hy  one  of  the  factors  of  the  divi- 
sor ;  then  divide  the  quotient,  thus  arising,  hy  a  second  factor, 
and  so  on,  till  every  factor  has  been  used  as  a  divisor:  the  last 
quotient  will  be  the  answer. 

EXAMPLES. 

Divide  the  following  numbers  by  the  factors  : 
1.  2322     by     6  =  2  x     3.      5.  1145592  by    72  =    8  X    9 


2.  37152  by  24  =  4  x     G. 

3.  19152  by  36  z=  G  X     G. 

4.  38592  by  48  =  4  X  12. 


G.  185760    by    96  =    8  X  12 

7.  115776    by    64=    8  x    8. 

8.  4G3104    by  144  =  12  x  12 


Note.  When  there  are  remainders,  after  division,  the  operation  la 
to  be  treated  as  one  of  Reduction. 


78  CONTE  ACTIONS 

9.  Divide  the  number  3671  by  3C  =  2  X  3  x  5. 

Analysis. — Dividing  3671  by  2,  yve  have  a  quotient  1S35,  and  a 
remainder,  1.     Alter  the  second 

division,  we  have  a  quotient  611,  operation. 

and  a  remainder,  2  ;  and  after  the  2)3671 

third  division,   the  quotient  122,  3)1835     .     .  1. 

and  the  remainder,  1.     Now.  it  is  5)611     .     .  2. 

plain,  from  the  first  analysis,  that,  122     ..   1. 

1.  The  unit  of  the  first  quotient  1X3  +  2=    3  +  2=    5; 

is  as  many  times  greater  than  the      5X2  +  1  =  10  +  1  =  11  rem. 
unit  of  the  dividend,  as  the  divi-  Ans.   122^^. 

sor  is  times  greater  than  1  ;   and 
similarly  for  all  the  following  quotients. 

2.  The  unit  of  the  first  remainder  is  the  same  as  the  unit  of  the 
dividend  ;  and  the  unit  of  any  remainder  is  the  same  as  that  of  the 
corresponding  dividend.  , 

3.  The  unit  of  any  dividend  is  reduced  to  that  of  the  preceding 
dividend,  by  multiplying  it  by  the  preceding  divisor. 

Hence,  to  find  the  remainder  in  imits  of  the  given  dividend,  is 
simply  a  case  of  reduction  in  which  the  divisors  denote  the 
units  of  the  scale  :  therefore, 

I.  Multiphj  the  last  remainder  hy  the  last  divisor  hut  one,  and 
add  in  the  preceding  remainder. 

II.  Multiphj  this  residt  hy  the  next  preceding  divisor,  and  add 
in  the  remainder,  and  so  on,  till  you  reach  the  unit  of  the  divi- 
dend. 

Divide  the  following  numbers  by  the  factors,  and  find  the 
remainders : 

1.  41G705  by       315  -    7x9x5. 

2.  80410G  by       462  =    S  x  2  x  7  x  11. 

3.  756807  by  3456  =  4  X  8  x  9  x  12. 

4.  8741659  by   105  =:  3  x  5  x  7. 

5.  917043  by   385  =  5  x  7  x  11. 

6.  4704967  by     1155  =  11x7x5x3. 

7.  71874607  bv     7560  =    8x7x9x5x3. 


IN    DIVISION.  79 

71.    When  the  divisor  is  10,  100,  1000,  S^c. 

1.  Divide  3278  by  1000  =  10  x  10  x  10. 

Analysis. — We  divide  3278  by  10,  by  operation. 

simply  cutting  off  8,  giving  327  tens  and  10)327|8 

8  units   remainder.     We  again  divide  by  10)32|7     .     8  rem. 

10,  by  cutting  ofT  the  7,  giving  32  hun-  10)3|2  .     .     7  rem. 

dreds  and  7  tens   remainder.     We   again  3      .     .     2  rem 

divide  by  !0  by  cutting  ofF  tlie   2,  giving  ^^-^^  Ans. 
a  quotient  of  3  tliousands  and  2  Imndreds 

remainder.     The  quotient  then  is  3,  and  a  remainder  of  2  hundreds 
7  tens  and  8  units,  or  278. 

Rule. —  Cut  off  from  the  right  of  the  dividend  as  many 
fgurcs  as  there  are  ciphers  in  the  divisor,  considering  the  figures 
at  the  left  the  quotient,  and  those  at  the  right  the  remainder. 

72.  When  any  divisor  contains  significant  figures  with  one  or 
more  ciphers  at  the  right  hand. 

2.  Divide  87589G  by  32000. 

Analysis. — The  divisor  32000  =  32  x  1000.  operation. 

Dividing  by  1000  gives  a  quotient  875,  and  896  32l000)875l896(27 
remainder.     Then  dividing  by  32  gives  a  quo-  64 

tient   27,  and  11  remainder,  Vi^hicli   gives  the  235 

result  271^11}.     Hence,  224 


Rule. —  Cut  off,  by  a  line,  the  ciphers  from      ^  ,  ^  „  „ 

**^  ^115.  27   A     — . 

the  right  of  the  divisor,  and  an  equal  num-  '      3 2000- 

ber  of  fgurcs  from  the  right  of  the  dividend :  divide  the  remain' 

ing  figures  of  the  dividend  by  ilie  remaining  figures  of  the  divisoVy 

and  the  remainder^  if  any,  with  the  figures  cut  off  from  the  divi' 

dend  annexed,  will  form  the  true  remainder. 

EXAMPLES. 

Divide  the  followinjr  numbers  : 


1.  1972G54  by  420000. 

2.  1752000  by  12000. 

3.  73199006  by  801400. 


4.  11428729800  by  72000. 

5.  3G981400  by  14G000. 

6.  141614398  by  63000. 


71.  How  do  you  divide  when  the  divisor  is  1,  with  ciphers  annexed  1 

72.  How  do  you  divide  when  the    divisor   contains    significant  figures, 
with  ciohers  annexed^     How  do  vou  find  the  true  remainder  1 


80  APFLICATiO.N.S 

78.    When  the  divisor  contains  a  fraction. 

1.  Divide  856  by  4i  operation. 

-Analysis. — There  are  5  filthy  in  1  ;  hence,    21=3x7    3)4280 
in  4  there  are  20   fifths;  therefore,  4^  =  21  7)142(i  -2 

fifths.     In  the  dividend  856,  there  are  5  times  203  -5 

as  many  fifths  as  units  1  ;  that  is,  4280  fifths  ;  Ans.  203t4. 

therefore,  the  quotient  is  4280  divided  by  21, 
equal  203-^|^.     Hence,  when  the  divisor  contains  a  fractional  part, 

Reduce  the  divisor  and  dividend  to  the  fractional  unit  of  the 
divisor,  and  then  divide  as  in  integral  numbers. 

Find  the  quotients  in  the  following  examples  : 


1. 

3245-^ 

-16^. 

5. 

87317^  9|. 

2. 

47804-^ 

-15}. 

6. 

87906  ^12f 

3. 

870631  -f- 

-14i. 

7. 

95675  ^  15f. 

4. 

37214-^ 

-511. 

8. 

71096 -M7f. 

APPLICATIONS    IN    MULTIPLICATION. 

74.  The  analysis  of  a  practical  question,  in  Multiplication, 
requires  that  the  multiplier  be  an  abstract  number ;  and  then 
the  unit  of  the  product  will  be  the  same  as  the  unit  of  the  mul- 
tiplicand. 

75.  To  find  the  cost  of  several  things,  lohen  we  kiwio  the 
price  of  one  and  the  numher  of  things  : 

1.  What  will  six  yards  of  cloth  cost  at  8  dollars  a  yard  ? 

Analysis. — Six  yards  of  cloth  will  cost  6  times  as  much  as  1  yard. 
Since  1  yard  of  cloth  cost  8  dollars,  6  yards  Avill  cost  6  times  8  dol- 
lars, which  are  48  dollars  ;  therefore.  6  yards  of  cloth  at  8  dollars  a 
yard,  will  cost  48  dollars  :  hence, 

TJie  cost  of  any  numher  of  things  is  equal  to  the  price  of  a 
single  thing  midtiplied  by  the  nu7nber  of  things. 

But  we  have  seen  that  the  product  of  two  numbers  will  be 
the  same,  (that  is,  will  contain  the  same  number  of  units)  which- 


73.  How  do  you  divide  when  the  divisor  contains  a  fraction  ! 

74.  What  docs  the  analysis  of  a  practical  question  lequire! 

75.  How  do  you  find  the  cost  of  a  single  thing  1     How  is  it  done  in 
practice  1 


ATPLICATIONS.  81 

ever  be  talcen  for  the  multiplicand  (Art,  50).  Hence,  in  prac- 
tice, we  may  multiply  the  two  factoi-s  together,  taking  either  for 
the  muhi2)lier,  and  then  assign  the  2>^'oper  unit  to  the  product. 
We  generally  take  the  less  number  for  the  multiplier. 

76.  To  find  the  cost  when  the  price  is  an  aliquot  part  of  a  dollar  : 
1.  Find  the  cost  of  45  bushels  of  apples,  at  25  cents  a  bushel. 

Analysis. — If  the  price  were  1  dollar  a  bushel,         operation. 
the  cost  would  be  as  many  dollars  as  there  aie  4)45,00 

bushels.     But  the  price  is  25  cents  =  ^  of  a  dol-  $11,25 

]ar;  hence,  the  cost  will  be   one-fourth   as   many 
dollars  as  there  are  bushels ;  that  is,  as  many  dollars  as  4  is  contained 
times  in  45  =  11  dollars  and  1  dollar  over.     This  is  reduced  to  cents 
by  adding  two  ciphers  j  then  dividing  again  by  4,  we  have  the  entire 
cost :  hence. 

Tithe  such  a  part  of  the  number  which  denotes  the  amount  (f 
the  commodity,  as  the  price  is  of  1  dollar :  the  result  ivill  be  the 
cost  in  dollars. 

EXAMPLES. 

1.  What  would  be  the  cost  of  284  bushels  of  potatoes,  at  50 
cents  a  bushel  ? 

2.  At  331  cents  a  gallon,  what  will  51  gallons  of  molasses  cost? 

3.  What  cost  112  yards  of  calico,  at  12^  cents  a  yard  ? 

4.  If  a  ponnd  of  butter  cost  20  cts.,  what  will  175  pounds  cost  ? 

5.  What  will  576  bushels  of     2)S576  cost  at  1  dollar  a  bushel. 

wheat  cost,  at  $1,50  a  bushel  ?        J^     ''     j^^e^its 

i?>64     "     Sfl,50  " 

6.  What  will  it  cost  to  dig  a  ditch  129  rods  long,  at  §1,331 
a  rod  ? 

7.  At  $1,25  a  barrel,  what  will  96  barrels  of  apples  cost? 

8.  What  will  5  pieces  of  cloth  cost,  each  piece  containing  25 
yards,  at  $1,20  a  yard  ? 

77.  To  find  the  cost  of  articles  sold  by  the  100  or  1000. 

1.  What  will  544  feet  of  lumber  cost  at  2  dollars  per  100  ? 

76.  How  do  you  find  the  cost,  when  the  price  is  an  aliquot  part  of  a  dollar  1 

77.  How  do  you  find  the  cost  of  articles  sold  by  the  hufidred  or  thou- 
sand 1 

4* 


82  APPLICATIONS^  S. 

Analysis. — At  2  dollars  a  foot,  the  cost  would  be  544  x  2  =  1088 
dollars  :  but  as  2  dollars  is  the  price  of  100  feet,  it  foJlows  that  1088 
dollars  is  100  times  the  cost  of  the  lumber;  therefore,  if  ^ye  divide 
1088  dollars  by  100  (which  is  done  by  cutting  off  two  of  the  right 
hand  figures,  Art.  71),  we  obtain  the  cost. 

Note. — Had  the  price  been  so  much  per  1000,  we  should  have 
divided  by  1000  :  hence, 

Multiply  the  quantity  hy  the  number  denoting  the  price  ;  if  the 
price  he  by  the  100,  cut  of  two  figures  on  the  right  hand  of  the 
product ;  if  by  the  1000,  cut  off  three,  and  the  remaining  fig- 
ures will  be  the  answer  in  the  same  denomination  as  the  price, 
tvhich,  if  cents  or  mills,  may  be  reduced  to  dollars. 

EXAMPLES. 

1.  What  will  be  the  cost  of  3742  feet  of  timber  at  $3,25 
per  100  ? 

2.  At  $12,50  per  1000,  what  will  5400  feet  of  boards  cost? 

3.  Richard  Ames,  Bought  of  John  Maple. 
1275  feet  of  boards  at    $9,00  per  1000 


3720 

u 

15,25 

it 

715 

scantling 

8,75 

u 

1200 

timber 

12,0G 

u 

2550 

lathing 

75 

lOi 

965 

plank 

1,121 

(< 

Received  payment,  John  Maple. 

*76.    To  find  the  cost  of  articles  sold  by  the  ton. 
What  is  the  cost  of  640  pounds  of  hay  at  $11,50  per  ton? 

Analysis. — Since  there  arc  2000  operation. 

pounds  in  a  ton,  the  cost  of  1000  2)$1].50 

l)ouiids  will  be  half  as  much  a.s  of  5.75    price  of  1000  lbs 

1  ton  :  viz.,  $5,75.     Mulliply  this  640 

by  the  number  of  pounds  (640).  and  23000 

cut  off  three  places  from  the  right  3450 

(Art.  71),   in  addition  to  the  two  $3^68000 

places  cut  off  for  cents  ;  hence, 

*76.  How  do  you  find  the  cost  of  articles  sold  by  the  toni 


APPLICATIONS.  83 

MxdiqAy  one-half  the  price  of  a  ion  by  the  number  of  pounds, 
and  cut  ojf  three  figures  from  the  right  hand  of  the  product.  The 
remaining  figures  will  be  the  ansiver  in  the  same  denomination  as 
the  price  of  a  ton. 

EXAMPLES. 

1.  What  will  be  the  cost  of  1575  pounds  of  plaster  at  $3,84 
per  ton  ? 

2.  At  $7,37-1-  a  ton,  what  will  3496  pounds  of  coal  cost  ? 

3.  What  will  1260  pounds  of  hay  cost  at  $9,40  per  ton  ?  at 
$10,25  ?  at  $14,60  ? 

4.  What  will  be  the  cost  of  transportation  of  5482  pounds 
of  iron  from  Pittsburgh  to  New  York  at  $6,65  per  ton  ? 

APPLICATIONS    IN   DIVISION. 

*77.  Abstractly,  the  object  of  division  is  to  find  from  two  given 
numbers  a  third,  which,  multiplied  by  the  first,  will  produce 
the  second.     Practically,  it  has  three  objects  : 

1.  Given  the  number  of  things  and  their  cost,  to  find  the 
price  of  one  thing. 

2.  Given  the  cost  of  a  number  of  things  and  the  price  of 
one  thing,  to  find  the  number  of  things. 

3.  To  divide  any  number  of  things  into  a  given  number  ot 
equal  parts. 

Analysis. — Consider  the  number  denoting  cost  or  price  as  abstract; 
then  make  the  division  and  assign  the  proper  unit  to  the  quotient. 
Hence,  we  have  the  following 

RULES 

I.  Divide  the  number  denoting  the  cost  ly  the  number  of 
things  :  the  quotient  will  be  the  price  of  one. 

II.  Divide  the  number  denoting  the  cost  hy  the  price  of  one  : 
the  quotient  will  be  the  number  of  things. 

III.  Divide  the  whole  number  of  things  by  the  number  of 
2)arls  into  which  they  are  to  be  divided:  the  quotient  will  be  the 
number  in  each  part. 


^77.  What  is  the  object  of  Division  abstractly?     How  many  objects  haa 
it  practically  ^     Name  the  objects.     Give  the  rules  for  the  three  cases. 


84* 


PRACTICE. 


PRACTICE. 

77*.  Practice  is  an  easy  method  of  applying  the  rules  oS 
Anthraetic  to  questions  which  occur  in  trade  and  business. 

An  aliquot  part  of  a  number  is  an  exact  part :  hence,  if 
a  number  be  divided  by  an  aliquot  part,  the  quotient  will  be  an 
integral  number. 

TABLE    OF   ALIQUOT   PARTS. 


Cts. 

Parts 
of$l. 

Parts  of£l. 

Parts  of  1 
f<hilling. 

Mo. 

Parts  of 
a  year. 

Days. 

Parts  of 
1  month. 

50   = 

h 

10s.        =^ 

6  d.=i 

6  = 

i 

15   =- 

i 

33^= 

i 

Gs.  8(1.=^ 

4  J.=l 

4  = 

i 

10    = 

i 

25   =- 

i 

5s.        =1 

3  d.=\ 

3  = 

i 

H  = 

i 

20   = 

i 

4s.       =1 

2  d.=l 

2  = 

i 

6    = 

i 

12i  = 

i 

3s.  4rf.=i 

lirf.=i 

1  = 

tV 

5    = 

1 

6 

6i  = 

iV 

2s.  6d.=\ 

1  d.  =  ^^ 

or  ^  of 

3    = 

iV 

5    == 

^ 

ls.8d.=^\ 

3mo. 

EXAMPLES. 

1.  What  is  the  cost  of  37G  yards  of  cloth  at  $1,75  =  1^ 
dollars  per  yai'd? 

Analysis. — At  $1  a  yard  it  would  cost  $376  ;  At  Si,     —       $376 

at   50  cents  =  S^;  a  yard,  it  would  cost  $188;  At  50cts.  =  Si  =  188 

at    25   cents  =  $|-  a   yard,  it  would  cost    $94  ;  At  25cts.=t^=   94 

hence,  at  $1,75  =  $1|,  it  will  cost  $658.  total  cost,     $658 

2.  What  is  the  cost  of  196  yards  of  cotton  at  9  c?.  =  |s.  per 
yard?  Ans.  £7  7s. 

3.  What  is  the  cost  of  425  yards  of  tape  at  lie?,  per  yard  ? 

Jiis.  £2  13s.  lirf. 

4.  What  is  the  cost  of  475  yards  of  tape  at  l^c?.  per  yard  ? 

Ans.  £2  9s.  o^d, 


PEAOTICE.  85* 

5.  "What  is  the  cost  of  354  yards  of  cord  at  l^d.  per  yard  ? 

Ans.  £1  16s.  lO^d. 

6.  At  121  cents  =  ||  a  yard,  Avhat  will  be  the  cost  of  4756 
yards  of  bleached  shirting  ?  A7is.  $594,50. 

7.  At  2s.  6d.  =  £J  per  pair,  what  will  be  the  cost  of  3754 
pairs  of  gloves  ?  -  Ans.  £469  5s. 

8.  If  wlieat  is  3s.  Gd.  a  bushel,  what  will  be  the  cost  5320 
bushels?  Aiis.  £931. 

9.  If  bi-oadcloth  costs  £1  7s.  a  yard,  what  will  be   tlie   cost 
of  435  yards?  Ans.  £587  5s. 

10.  If  linen  is  2^.  Gd.  =  2^s.  a  yard,  what  will  be   the  cost 
of  660  yards  ?  Ans.  £82  10s. 

11.  What  will  be  the  cost  of  AOlbs.  of  soap,  if  1  pound  costs 
6f  cents?  ylns.  $2,70. 

12.  If  a  yard  of  twisted   cord   costs  21  cents,  what  will  be 
the  cost  of  140  yards  ?  Ans.  $3,15. 

13.  If  one  bushel  of  apples  cost  62^  cents,  what  will  be   the 
cost  of  876  bushels  ?  Ans.  $547,50. 

14.  What  will  be  the  cost  of  1000  quills,  if  every  5  quills 
cost  11  cents  ?  Ans.  $3,00. 

15.  If  1  yard  of  extra-superfine  cloth  costs  |91,  what  will 
be  the  cost  of  851  yards  ?  ^,i5.  f  812,25. 

16.  What  will   be   the  cost  of  1848   yards  of  linen   cambric 
at  87^  cents  a  yard?  Ans.  $1617. 

17.  If  one    yard    of   broadcloth    cost    $4,871   cents  =  $41, 
what  will  be  the  cost  of  696  yards  ?  Ans.  $3393. 

18.  If  the  price  of    1    yard  of  cloth  is   $4,75  z=:  $4f,  what 
will  be  the  cost  of  281  yards  ?  Ans.  $135,375. 

19.  If  1  quart  of  oil  costs  14^  cents,  what  will  be  the  cost 
of  Ihhd.  2gaL  3qts.  ?  Ans.  $38,135. 

20.  What  will  be  the  cost  of  350  bushels  of  potatoes,  at  3s. 
Qd.  =  3^s.  a  bushel  ?  Ans.  £61  5s. 


84  LONGITUDE   AND   TIME. 

LONGITUDE    AND    TIME. 

78.  The  equator  of  the  earth,  like  other  circles,  is  divided 
into  360°,  which  are  called  degrees  of  longitude. 

79.  The  sun  apparently  goes  round  the  earth  once  in  24 
hours.     This  time  is  called  a  day. 

Hence,  in  24  hours,  the  sun  apparently  passes  over  3G0^  of 
longitude;  and  in  1  hour  over  360'^-i-24  =  15''. 

80.  Since  the  sun,  in  passing  over  15°  of  longitude,  requires 
1  hour  or  60  minutes  of  time,  1°  will  require  60  minutes 
-r-lo  =  4  minutes  of  time;  and  1'  of  longitude  will  require 
one-sixtieth  of  4  minutes,  which  is  4  seconds  of  time  :  hence, 

15°  of  longitude  require  1  hour. 
1°  of  longitude  requires  4  minutes. 
1'  of  longitude  requires  4  seconds. 
Hence,  we  see  that, 

1.  If  the  degrees  of  longitude  be  multiplied  by  4,  the  2>'>'oduct 
will  be  the  corresponding  time  in  minutes. 

2.  If  the  minutes  in  longitude  be  multiplied  by  4,  the  jyroduct 
will  be  the  corresp)onding  time  in  seconds. 

1.  What  is  the  time  in  hours,  minutes  and  seconds  in  56°  47'  ? 

Analysis. — First  reduce  the  de-  operation. 

grees  to  hours  and  minutes ;  then  m.  hr.  m.  sec. 

reduce  the  minutes  to  minutes  and     56°  x  4  =  224         =3     44 
seconds,  and  take  the  .svim.  47' X  4  =  18S  sec.  =  3  8 

3     47  8 

81.  When  the  sun  is  on  the  meridian  of  any  place,  it  is  12 
o'clock,  or  noon,  at  that  place. 

78.  How  is  the  equator  of  the  earth  supposed  to  be  divided  i 

79.  How  does  the  sun  appear  to  move  '  "What  is  a  day  1  How  far 
docs  the  sun  appear  to  move  in  1  hour ! 

80.  How  do  you  reduce  degrees  of  longitude  to  time  1  How  do  you 
reduce  minutes  of  longitude  to  time  ? 

8L  What  is  the  hour  when  the  sun  is  on  tlie  meridian  1  When  the 
eun  is  on  the  meridian  of  any  place,  how  will  the  time  be  for  all  places 
east  1  How  for  all  places  west  1  If  you  have  the  difference  of  time,  how 
do  you  find  the  time  1 


LONGITUDK    AND   TIME.  85 

Now,  as  the  sun  apparently  goes  from  east  to  west,  at  the 
instant  of  noon,  at  one  place,  it  will  be  past  noon  for  all  places 
at  the  east,  and  before  noon  for  all  places  at  the  west. 

If  then,  Ave  find  the  difference  of  time  between  two  places 
and  know  the  exact  time  at  one  of  them,  the  corresponding 
time  at  the  other  will  be  found  by  adding  their  difference,  if 
that  other  be  east,  or  by  subtracting  \i,\i  west. 

82.    2^0  reduce  time  to  degrees  and  minutes  of  longitude. 

1.  The  difference  of  time  between  Boston  and  New  Orleans 
is  1  hour  11  minutes  and  48  seconds  :  what  is  the  difference  of 
lonfritude  ? 

Analysis. — Since  1  hour  corresponds  operation. 

to   15°  of  longitude,  there  will   be   as  15°  x  1  =  15° 

many  times   15°  as   thei-e   are   hours:  11   -i- 4=    2°  45' 

Since    1°  corresponds  to   4   minutes  of  48  -i- 4=  12' 

time,  there  will  be  as  many  degrees  as  Diff.    17°  57' 

4  is  contained    times   in  the  minutes  : 

Since   1'  corresponds  to   4   seconds  of  time,  there  will  be  as  many 
minutes  as  4  is  contained  times  in  the  seconds  :  hence, 

1.  Multiply  15°  by  the  number  of  hours,  and  the  'product  will 
he  degrees  of  longitude : 

2.  Divide  the  minutes  by  4,  and  the  quotient  will  be  degrees 
and  minutes  of  longitude : 

3.  Divide  the  seconds  by  4,  and  the  quotient  ivill  be  minutes 
and  secunds  of  longitude.  The  sum  of  these  results  will  be  the 
difference  of  longitude. 

EXAIMPLES. 

1.  The  longitude  of  Albany  is  73°  42'  west,  and  that  of 
Buffalo  78°  55'  west :  what  is  the  difference  of  longitude  and 
what  the  difference  of  time  ? 

2.  The  longitude  of  New  York  is  74°  1'  west,  and  that  of 
Springfield,  Illinois,  89°  33'  west:  what  would  be  the  time  at 
New  York  when  it  is  12  M.  at  Springfield  ? 

82.  How  do  you  reduce  time  to  degrees  and  minutes  of  longitude  '^ 


86  APPLICATIONS. 

3.  When  it  is  12  M.  at  New  York,  it  is  11  o'clock  6  minutea 
and  28  seconds  at  Cincinnati :  wliat  is  their  difference  of  lon- 
gitude ? 

4.  The  longitude  of  Philadelphia  is  75°  10'  Avest,  and  that 
of  New  York  74°  1'  west:  what  is  the  difference  of  time  be- 
tween these  two  places  ? 

5.  Washington  is  in  longitude  77°  2'  west,  New  Orleans  in 
89°  2'  west.  When  it  is  9  o'clock  A.  M.  at  Washing-ton,  what 
is  the  time  at  New  Orleans  ? 

6.  If  the  difference  of  time  between  two  places  be  42/».z. 
IGiCc,  what  is  the  difference  of  longitude? 

7.  What  is  the  difference  of  longitude  between  two  places  if 
the  difference  of  time  is  2//.  20»i/.  44sec,  ? 

8.  The  longitude  of  St,  Louis  is  90°  15'  west;  a  person  at 
that  place  observed  an  eclipse  of  the  moon  at  lOA.  40;/i?.  P.  M. ; 
another  person,  in  a  neighboring  state,  observed  the  same  eclipse 
22/m/.  125fr.  earlier:  what  was  the  longitude  of  the  latterplace, 
and  the  time  of  observation  ? 

9.  If  the  difference  of  time  between  London  and  Ore"-on 
Citj  is  8  hours,  what  is  the  difference  in  longitude  ? 

10.  The  difference  of  longitude  between  St.  Louis  and  New 
l^ork  is  15°  35'.  In  travelling  from  New  York  to  St.  Louis 
will  a  watch,  keeping  accurate  time,  be  fast  or  slow  at  St.  Louis, 
and  how  much  ? 


APPLICATIONS    OF   THE   PRECEDING   RULES. 

1.  What  will  it  cost  to  build  a  wall  9G  rods  long,  at  $1,33}  a 
rod  ? 

2.  A  farmer  wishes  to  put   lOGGiws/^  Iph.  of  potatoes  into 
474  barrels,  how  much  nuist  he  put  into  each  barrel  ? 

3.  At  $4,32  a  yard,  what  will  121  yards  of  cloth  cost  ? 

4.  How  many  barrels  of  apples,  each  containing  1\  bushels, 
can  I  buy  for  $3G,  at  45  cents  a  bushel  ? 

5.  The  quotient  arising  from  a  certain  division  is  123G  ;  the 
divisor  is  375,  and  the  remainder  184  :  what  is  the  dividend  1 


APPLICATIONS.  87 

G.  The  Croton  Water  Works  of  New  York  are  capable  of 
discharging  60000000  gallons  of  water  every  24  hours  :  what 
would  be  the  average  amount  per  minute  ? 

7.  The  population  of  the  United  States,  in  1850,  was  23191876. 
It  has  been  estimated  that  1  person  in  every  400  dies  from 
intemperance  :  how  many  deaths  then  may  be  attributed  to  this 
cause,  in  the  United  States,  during  that  year  ? 

8.  At  the  rate  of  45  miles  an  hour,  how  long  would  it  take  a 
railroad  car  to  pass  around  the  globe,  a  distance  of  25000  miles  ? 

9.  If  a  quantity  of  provisions  will  last  25  men  2mo.  owl:.  Gda., 
how  long  will  it  last  10  men  ? 

10.  If  a  man's  salary  is  $1200  a  year,  and  his  expenses  are 
$640  annually,  how  many  years  will  it  be  before  he  will  save 
$6720  ? 

11.  How  long  will  it  take  to  count  20  millions  at  the  rate  of 
80  per  minute  ? 

12.  If  3160  barrels  of  pork  cost  $47400,  how  many  barrels 
can  be  bought  for  $11475? 

13.  What  will  be  the  cost  of  6  firkins  of  butter,  each  con- 
taining 96  pounds,  at  12^  cents  a  pound  ? 

14.  What  will  1000  quills  cost,  at  1  cent  a  piece  ? 

15.  What  will  be  the  cost  of  851-  yards  of  cloth,  at  $9i  a 
yard? 

16.  What  will  be  the  cost  of  l/thJ.  2gcd.  oqt.  of  brandy,  at 
56^  cents  a  quart  ? 

17.  What  will  be  the  cost  of  196  yards  of  cotton  goods,  at 
Is.  6d.  per  yard  ? 

18.  At  2s.  8d.  per  bushel,  what  will  1246  bushels  of  oats  cost  ? 

19.  If  112Z5.  of  cheese  cost  £2    16s.,   what    is    that    per 

pound  ? 

20.  What  will  be  the  cost  of  1426  pounds  of  hay,  at  $9,75 

per  ton  ? 

21.  How  much  must  I  pay  for  the  transportation  of  3840 
pounds  of  iron,  from  Albany  to  Buflxilo,  at  $4,50  per  ton  ? 

22.  Bought  124  barrels  of  potatoes,  each  containing  2\  bush 
els,  at  33i  cents  a  bushel :  what  is  the  whole  cost  ? 


88  APl'I.ICATIONS. 

23.  If  fifteen  hundred  tons  of  coal  cost  $11812,50,  what  will 
one  ton  cost  ? 

24.  If  789  pounds  of  leatlier  cost  $142,02,  what  is  that  per  lb.  ? 
2a.  There  are   three  numbers,  whose  continued  product  is 

10)200;  one  of  the  numbers  is   25;  another  18:  what  is  the 
third  number? 

2G.  If  Idwt.  of  gold  is  worth  92  cents,  what  would  be  the 
weight  of  $10059,28  in  gold? 

27.  A  man  sold  his  house  and  lot  for  $4200,  and  took  his 
pay  in  railroad  stock,  at  84  dollars  a  share ;  how  many  shares 
did  he  receive  ? 

28.  A  person  bought  640  acres  of  land,  at  15  dollars  an 
acre.  He  afterwards  sold  160  acres  at  20  dollars  an  acre  ; 
240  acres  at  18  dollai's,  and  for  the  remainder  he  received 
84560.  What  was  his  entire  gain,  and  what  did  he  receive 
per  acre  on  the  last  sale  ? 

29.  A  piece  of  ground  60  feet  long  and  48  feet  wide  is  en- 
closed by  a  wall  12  feet  high  and  2^  feet  thick  :  how  many 
cubic  feet  in  the  wall  ? 

30.  What  will  be  the  cost  of  transportation,  from  Montreal 
to  Boston,  of  325640  feet  of  lumber  at  $2,37i  per  thousand  ? 

31.  Bought  684  pounds  of  hay,  at  $12.40  a  ton :  what  will  it 
cost  me  ? 

32.  At  $2,121  a  hundred,  what  will  786  feet  of  lumber  cost? 

33.  How  many  shingles  will  it  require  to  cover  the  roof  of  a 
building  40  feet  long  and  26  feet  wide,  witli  rafters  16  feet  long, 
allowing  one  shingle  to  cover  24  square  inches  ? 

34.  If  14/3.  %oz.  12dwt.  3ffr.  of  silver  be  made  into  9  tea-pots 
of  equal  weight,  what  will  be  the  weight  of  each? 

35.  A  man  bought  320  barrels  of  flour  for  $2688:  at  what 
rate  must  he  sell  it  to  gain  $1,60  on  each  barrel  ? 

36.  A  farmer  has  a  granary  containing  Ai[)bi(s/i.  Ipk.  2qt.  of 
wheat ;  he  wishes  to  put  it  into  182  bags:  how  much  must  he 
put  in  each  bag  ? 

37.  A  trader  bought  750  barrels  of  flour  for  which  he  paid 
$4875  ;  ho  sold  the  same  for  $7,25  a  barrel :  what  was  his 
profit  on  each  banei  ? 


AI'l'LICATIONS.  89 

38.  How  many  sheep,  at  $1,62J  a  head,  can  be  bought  for  $1 G9  ? 

39.  If  a  person  save  $6,87-l-  a  day,  how  long  will  it  take  hira 
to  save  $267,75  ? 

40.  How  many  canisters,  each  holding  3lb.  lOoz.,  can  be  filled 
from  a  chest  of  tea  containing  58lb. 

41.  In  26  hogsheads  the  leakage  has  reduced  the  whole 
amount  to  l35Sff(il.  2qt. ;  if  the  same  quantity  has  leaked  out  of 
each  hogshead,  how  much  still  remains  in  each  ? 

42.  The  number  of  college  libraries  in  the  United  States  in 
1850  was  213,  containing  942312  volumes  :  what  would  be  the 
average  number  of  volumes  to  each  ? 

43.  A  man  bought  a  piece  of  land  for  $3475,25,  and  sold  the 
same  for  $3801,65,  by  which  transaction  he  made  $3,40  an 
acre  :  how  many  acres  were  there  ? 

44.  The  whole  amount  of  gold  produced  in  California  in  the 
year  1855,  was  as  follows  :  $43313281,  sent  to  the  Atlantic 
States  ;  $6500000,  sent  directly  to  England  ;  and  $8500000 
retained  in  the  country.  In  1854,  the  total  product  of  gold  in 
California  was  $57715000  :  how  much  more  was  produced  in 
1855  than  in  1854  ? 

45.  If  the  forward  wheels  of  a  carriage  are  12  feet  in  cir- 
cumference, and  the  hind  wheels  16  feet  6  inches,  how  many 
more  times  will  the  forward  wheels  turn  round  than  the  hind 
wheels,  in  runninfj  a  distance  of  264  miles  ? 

46.  If  a  certain  township  is  9  miles  long,  41  miles  wide,  how 
many  farms  of  192  acres  each  does  it  contain  ? 

47.  The  total  number  of  land  warrants  issued  during  the 
year  ending  Sept.  30th,  1855,  was  34337,  embracing  4093850 
acres  of  land :  what  was  the  average  number  of  acres  to  each 
warrant  ? 

48.  The  amount  of  foreign  imports  brought  into  the  Uniled 
States  during  the  fiscal  year  of  1855  Avas  $261382960;  during 
the  year  1854  it  was  $305780253  :  how  much  was  the  decrease  ? 

49.  The  longitude  of  Philadelphia  is  75"  10',  and  that  of 
New  Orleans  89°  2' ;  when  it  is  12  M.  at  Philadelphia,  what  is 
the  time  at  New  Orleans  .'' 


90  APPLICATIONS. 

50.  The  sun  passes  the  meridian  at  12  M.,  the  moon  at 
%hr.  30m.  P.  M.  :  what  is  the  difference  in  longitude  between 
the  sun  and  moon  ? 

51.  Two  persons,  A  and  B,  observed  an  eclipse  of  the  moon  ; 
A  observed  its  commencement  at  Qhr.  A2mi.  P.  INI. ;  B  was  in 
longitude  73°  20',  and  observed  its  commencement  23  minutes 
eai-lier  than  A:  what  was  A's  longitude,  and  B's  time  of 
observation  ? 

52.  If  in  11  piles  of  wood  there  are  120  cords,  7  cord  feet, 
5    cubic  feet,  how  much  is  there  in  each  pile  ? 

53.  If  IQavL  2q)\  lllb.  lOoz.  of  flour  be  put  into  9  barrels, 
how  much  will  each  barrel  contain  ? 

54.  A  miller  bought  a  quantity  of  wheat  for  $G25,40,  which 
he  floured  and  put  into  barrels  at  an  expense  of  Si  10,12^: 
what  profit  did  he  make  by  selling  it  for  $900  ? 

55.  America  was  discovered  Oct.  11th,  1492  :  liow  long  to 
the  commencement  of  the  Revolution,  April  19th,  1775  ? 

56.  From  a  hogshead  of  wine  a  merchant  draws  18  bottles, 
each  containing  Ijyt.  Zgills  ;  he  then  fills  three  6  gallon  demi- 
jons,  and  4  dozen  bottles  each  containing  2qt.  Iq^t.  Skills:  how 
much  remained  in  the  cask  ? 

57.  In  753 G89  yards,  how  many  degrees  and  statute  miles  ? 

58.  In  189mz.  3fur.  6rd.  \ft.  how  many  feet  ? 

59.  If  24  men  can  build  768  rods  of  wall  in  1  day,  how 
many  rods  can  48  men  build  in  9  days  ? 

60.  A  certain  number  increased  by  1764,  and  the  sum  mul- 
tiplied by  209,  gives  the  product  of  7913576 :  what  is  the 
number  ? 

61.  If  a  man  travel  146mi.  Ifur.  Ih-d.  14/L  in  5  days,  how 
much  is  that  for  each  day  ? 

62.  If  325  acres  of  land  cost  $17712,50,  how  many  acres 
can  be  bought  for  $545  ? 

63.  A  merchant  having  $324  wishes  to  purchase  an  equal 
number  of  yards  of  two  kinds  of  cloth ;  one  kind  was  worth 
4  dollars  a  yard,  the  other  was  worth  5  dollars  a  yard :  how 
many  yards  of  each  can  he  buy? 


APPLICATIONS.  91 

64.  From  one-fourth  of  a  piece  of  cloth  containing  QSyd.  3qr. 
a  tailo"  cut  5  suits  of  clothes  :  how  much  did  each  suit  contain  ? 

Go.  A  manufacturer  having  £5  lO.v,,  distributed  it  among  his 
laborers,  giving  every  man  18d.,  every  woman  12c/.,  and  every 
boy  lOd. ;  the  number  of  men,  women  and  boys,  were  equal : 
what  was  the  number  of  each  ? 

GG.  It  is  estimated  that  1  out  of  every  1585  persons  in  Great 
Britain  is  deaf  and  dumb.  The  population,  according  to  the 
census  of  1851,  was  2093G4G8  :  how  many  deaf  and  dumb 
persons  were  there  in  the  entire  population  ? 

67.  A  grocer  in  packing  6  dozen  dozen  eggs  broke  half  a 
dozen  dozen,  and  sold  the  remainder  for  1^-  cents  a  piece :  how 
much  did  he  receive  for  the  e2:2;s  ? 

68.  How  much  time  will  a  man  save  in  50  years,  beginning 
witli  a  leap  year,  l)y  rising  45  minutes  earlier  each  day  ? 

GD.  liichard  Roe  was  born  at  6  o'clock,  A.  M.,  June  24th, 
18-32  :  what  will  be  his  age  at  3  o'clock,  P.  M.,  on  the  10th  day 
of  January,  1858? 

70.  During  the  year  1855,  there  were  shipped  to  Great 
Britain,  from  the  United  States,  408434  barrels  of  tloiir  ;  2550092 
bushels  of  wheat;  1048540  bushels  of  corn.  Supposing  the 
flour  to  have  sold  for  $10,25  a  barrel,  the  wheat  for  $2,121  a 
bushel,  and  the  corn  for  $0,94  a  bushel,  what  was  the  value 
of  the  whole  ? 

71.  A  man  dying  without  making  a  will,  left  a  vridow  and 
4  children.  The  law  provides,  in  such  cases,  that  the  Avidow 
shall  receive  one-third  of  the  personal  propei'ty,  and  thai  the  re- 
mainder shall  be  equally  divided  among  the  children.  The 
estate  was  valued  as  follows  :  a  farm,  at  $5000 ;  5  horses,  at 
$85  each;  a  yoke  of  oxen,  for  $110;  25  cows,  at  $22  each; 
150  sheep,  at  $2  each  ;  some  lumber,  at  $45  ;  forming  utensils, 
at  $174;  household  furniture,  at  $450  ;  grain  and  hay,  at  $380  : 
what  was  the  share  of  the  widow  and  each  child  ? 

72.  The  amount  of  gold  coin  in  the  United  States  in  1855 
was  estimated  at  about  $241200000.  Adopting  the  same  ratio 
of  increase  as  from  1850  to  1855,  the  population  of  the  United 


92 


APPLICATIONS. 


States  in  1855  would  be  about  26800000.     In  an  equal  distri- 
biition  of  the  gold,  how  much  would  eacli  person  receive  ? 

73.  How  many  sliingles  will  it  take  to  cover  the  two  sides  of 
the  roof  of  a  building,  55  feet  long,  with  rafters  IGi  feet  in 
length,  allowing  each  shingle  to  be  15  inches  long  and"  4  inches 
wide,  and  to  lay  one-third  to  the  weather  ? 

74.  If  the  longitude  of  St.  Petersburgh  is  30°  45'  east,  and 
that  of  Washington  77°  2'  west,  what  is  the  difference  of  lon- 
gitude between  the  two  places,  and  the  difference  of  time  ? 

75.  When  it  is  6  o'clock,  A.  M.,  at  Washington,  what  is  the 
time  at  St.  Petersbursrh  ? 

76.  A  vessel  sails  from  New  York  to  Liverpool.  After  a 
number  of  days  the  captain,  by  taking  an  observation  of  the 
sun,  finds  that  his  chronometer,  which  gives  New  York  time, 
differs  \hr.  4:4-m.  from  the  time  at  the  place  of  observation.  If 
his  chronometer  shows  the  time  to  be  Shi:  12 mi.  P.  M.,  what  is 
the  correct  time,  at  the  place  of  observation,  and  how  far  is  he 
east  from  New  York  ? 

77.  A  cistern  containing  960  gallons,  has  two  pipes;  45 
gallons  ]-un  in  every  hour  by  one  pipe,  and  25  gallons  run  out 
by  the  other:  how  long  a  time  will  be  required  to  fdl  the 
cistern  ? 

78.  A  speculator  sold  840  bushels  of  wheat  for  $2180,  which 
was  S500  more  than  he  gave  for  it :  what  did  it  cost  him  a 
bushel  ? 

79.  The  wliole  number  of  gallons  of  rum  manufactured  in 
the  United  States  in  1850,  was  6500500  gallons  ;  if  it  be 
valued  at  50  cents  a  gallon,  how  many  schoolhouses  could  be 
built,  worth  $750  each,  with  the  proceeds  ? 

80.  A  farmer  sold  a  grocer  30  bushels  of  potatoes  at  37-i  cents 
a  bushel,  for  which  he  received  6  gallons  of  molasses  at  45 
cents  a  gallon  ;  GO  pounds  of  mackerel  at  6^-  cents  a  pound,  and 
tlie  remainder  in  sugar  at  10  cents  a  pound  :  how  many  pounds 
of  sugar  did  he  receive  ? 

81.  If  a  man  travel  12;///.  Sfu?:  20a/.  in  one  day,  how  long 
will  it  take  him  to  travel  174////.  \far.  at  the  same  rate  ? 


APPLICATIONS.  93 

82.  If  a  man  sell  '2har.  12^/al.  2qi.  of  beer  in  one  week,  Low 
much  will  he  sell  in  12  weeks  ? 

83.  A  hquoi-  merchant  had  ooO  pint  bottles,  400  quart  bottles, 
350  two  quart  bottles,  375  three  quart  bottles,  and  150  jugs, 
liolding  a  gallon  each  :  hew  many  barrels  of  wine  will  fill  them  ? 

84.  How  many  yards  of  carpeting,  one  yard  wide,  will  it 
take  to  cover  the  floors  cf  two  parlors,  each  18  feet  long,  and 
IG  feet  wide,  and  what  will  it  cost  at  $l,o3i  a  yard? 

85.  How  many  rolls  of  wall  paper,  each  10  yards  long  and 
2  feet  wide,  will  it  take  to  cover  the  sides  of  a  room  22  feet 
long  and  1 6  feet  wide  and  9  feet  high  ? 

8G.  Two  persons  ai'e  Imi.  Afiir.  20/ d.  apart,  and  are  travel- 
ling the  same  way.  The  hindmost  gains  upon  the  foremost 
5  rods  in  travelling  25  rods  :  how  far  must  he  travel  to  over- 
take the  foremost  ?  . 

87.  A  man  sold  500  bushels  of  wheat  at  $1,75  a  bushel,  and 
took  his  pay  in  sugar  at  5  cents  a  pound.  He  afterwards  sold 
one-half  of  it :  what  quantity  of  sugar  had  he  left  ? 

88.  A  man  bought  7  barrels  of  sugar  at  Sl2,87-l-  a  barrel; 
he  kept  two  barrels  for  his  own  use,  and  sold  the  remainder  for 
what  the  whole  cost  him  :  what  did  he  receive  per  barrel  ? 

89.  A  flour  merchant  bought  a  quantity  of  flour  for  $18750, 
and  sold  the  same  for  $26250,  by  which  he  gained  $3  a  barrel 
how  many  barrels  were  there  ? 

90.  Three  men  rented  a  farm  and  raised  d64:biish.  2jjI:  AqL 
of  grain,  which  was  to  be  divided  in  proportion  to  the  rent  paid 
by  each.  The  first  was  to  have  one-half  the  whole  ;  the  second 
one-third  the  remainder;  and  the  third  had  what  was  left :  how 
much  did  each  have  ? 

91.  A  vessel  in  longhude  70^  25'  east,  sails  105°  30'  56'- 
west,  then  40°  50'  east,  then  10'  5'  40"  west,  then  39°  11'  36" 
east ;  in  what  longitude  is  she  then,  and  how  many  days  will  it 
take  her  to  sail  to  longitude  77°  west,  if  she  sail  3°  20'  each 
da,}'  ? 

92.  A  privateer  took  a  prize  worth  $25000,  which  was 
divided  into  125  shares,  of  which  the  captain  took  12  shares ; 


94  APPLICATIONS. 

2  lieutenants,  eacli  5  shares  ;  6  midshipmen,  each  3  shares ; 
ajid  the  remainder  was  divided  equally  among  85  seamen :  how 
much  did  each  receive  ? 

93.  If  the  longitude  of  Boston  is  71°  4',  and  a  gentleman  in 
travelling  from  Boston  to  Chicao;o  iinds  that  his  watch  is  Ihr. 
5m.  44sec.  too  fast  by  the  time  of  the  latter  place,  what  is 
the  longitude  of  Chicago,  provided  his  watch  has  kept  time 
accurately  ? 

94.  What  time  would  it  be  in  Boston  when  it  was  8/ir.  21  mi. 
ZOsec,  A.  M.,  in  Chicago  ? 

95.  What  time  would  it  be  at  Chicago  when  it  was  12  M.  at 
Boston  ? 

96.  Two  places  lie  exactly  east  and  west  of  each  other,  and 
by  observation  it  is  found  that  the  sun  comes  to  the  meridian 
of  the  latter  place  1  hour  and  16  minutes  after  the  former: 
bow  far  apart  are  they  in  degrees  and  minutes  of  longitude  ? 

97.  In  12  bales  of  cloth,  each  bale  containing  16  pieces,  and 
each  piece  containing  20  ell  English,  how  many  yards  ? 

98.  How  many  eagles  can  be  made  from  2Ub.  4oz.  (5pwt. 
ISgr.  of  gold,  making  no  allowance  for  waste,  if  each  eagle 
weighs  llpwls.  9(/r.  ? 

99.  A  man  paid  $3284,82  for  some  wheat.  He  sold  740 
bushels  at  2  dollars  a  bushel ;  the  remainder  stood  him  in  $1,42 
a  bushel :  how  many  bushels  did  he  purchase  ? 

100.  A  speculator  gave  $8968  for  a  certain  number  of  barrels 
of  flour,  and  sold  a  part  of  it  for  $2618,  at  $7  a  barrel,  and  by 
so  doing  lost  $2^  on  each  barrel ;  for  how  much  must  he  sell 
the  remainder  to  gain  $1060  on  the  whole? 

101.  A  man  sold  105/1.  2R.  2QF.  of  land  ibr  as  many  dollars 
as  there  were  perches  of  land,  payable  in  instalments,  at  the  rate 
of  1  dolhu-  an  hour.  If  the  contract  was  closed  at  12  o'clock, 
M.,  April  1st,  1856,  what  length  of  time  will  be  allowed  the 
purchaser  to  pay  the  debt,  reckoning  365  days  6  hours  to  the 
year  ? 


PKOPERTIKS    OF    NUMBERS.  95 


PROPERTIES    OF    NUMBERS. 

PRIME    AND    COMPOSITE    NUMBERS. 

83.  An  Integral  Number  is  the  unit  1,  or  a  collection  of 
Buch  units. 

84.  One  number  is  said  to  be  divisible  by  another  when  the 
quotient  is  an  integral  number.  The  division  is  then  said  to  be 
exact. 

85.  A  Composite  Number  is  one  that  may  be  produced 
by  the  multiplication  of  two  or  more  numbers,  called  factors  ; 
thus,  30  =  2  X  3  X  5,  is  a  composite  number,  in  which  the 
factors  are  2,  3  and  5. 

Note  1. — A  composite  number  is  exactly  divisible  by  any  one  of 
its  factors. 

86.  A  Prime  Number  is  a  number  that  is  divisible  only 
by  itself  and  by  1 ;  thus,  1,  2,  3,  5,  7,  11,  13,  &c.,  are  prime 
numbers. 

87.  Two  numbers  are  said  to  be  prime  to  each  other  when 
they  have  no  common  factor ;  thus,  4  and  9  are  2^r//?ie  to  each 
other,  though  both  are  composite  numbers. 

88.  Any  number  (prime  or  composite),  as  26,  may  be  put 
under  the  form,  1  X  26;  hence,  every  number  is  divisible  by 
itself  and  by  1,  and  therefore,  these  are  not  reckoned  among 
i\iQ  factors  or  divi'sos  either  of  prime  or  composite  numbers. 

83.  What  is  an  integral  number  1 

84.  When  is  one  number  said  to  be  divisible  by  another]  How  is  the 
division  then  said  to  be  1 

85.  What  is  a  composite  niamber  1  Bv  what  is  a  composite  numbej 
always  divisible  1 

86.  What  is  a  prime  number  '; 

87.  When  are  two  numbers  prime  to  each  other  1 

88.  What  numbers  are  not  reckoned  among  the  divisors  of  prime  or 
composite  numbers  1 


96  PKOPEKTIES    OF    NrHIBERS. 

89.  Every  factor  of  a  composite  number  is  a  divisor,  and 
is  either  prime  or  composite  ;  and,  since  every  composite  factor 
may  be  again  divided,  it  follows  that 

Every  number  is  equal  to  the  product  of  all  its  prime  factors. 

For  example,  24  =  3  X  8 ;  but  8  is  a  composite  number  of 
which  the  factors  are  2  and  4 ;  and  4  is  a  composite  number  of 
which  the  factors  are  2  and  2 ;  hence, 

24  =  3x     8  =  3x2x4=33x2x2x2;  and 
60  =  5x12  =  5x3x4  =  5x3x2x2. 

Hence,  to  find  the  prime  factors  of  any  number  : 
Divide  the  number  by  any  prime  member  that  will  exactly 
divide  it:  then  divide  the  quotient  by  any  prime  number  that 
will  exactly  divide  it^  and  so  on,  till  a  quotient  is  found  which 
is  prime ;  the  several  divisors  and  the  last  quotient  will  be  the 
prime  factors  of  the  given  number. 

Note. — It  is  most  convenient,  in  practice,  to  use  at  each  division 
the  least  prime  number  that  is  a  divisor. 

1.  What  are  the  prime  factors  of  105  ? 

Analysis. — Three  being  the  least  divisor  that  operation. 
is  a  prime  number,  Ave  divide  by  it,  giving  the  3)105 

quotient  35  ;  then  5  is  the  least  prime  divisor  5)35 

of  this  quotient :   hence,   3,   5   and   7  are  the  7 

prime  factors  of  105. 

EXAMPLES. 

1.  What  are  the  prime  factors  of  9?  10?  12?  14?  IC?  1«? 
24?  27?  28? 

2.  What  are  the  prime  factors  of  30?  22?  32?  BG?  38?  40? 
45?  49? 

3.  What  are  the  prime  factors  of  50  ?  56?  58?  60?  64  66? 
68?  70?  72? 

4.  What  are  the  prime  foctors  of  7G  ?  78?  80?  82?  84? 
86?  88?  90? 

89.  To  what  product  is  every  number  equal  ]  How  do  you  find  the 
prime  factors  of  any  number  1 


PROPERTIES    OF    NUMBERS.  97 

5.  What  are  the  prime  factors  of  92?  94?  96?  98?  99? 
100?  102?  104? 

G.  What  are  the  prime  fxctors  of  105?  lOG  ?  108?  110? 
115?  116?  120?  125? 

7.  What  are  the  prime  foctors  of  302  ?  305  ?  604  ?  875  ? 
975  ?  055  ? 

Note. — The  prime  factors,  "when  the  numbers  are  small,  may 
generally  be  seen  by  inspection.  The  teacher  can  easily  increase 
the  number  of   examples. 

90.  When  there  are  several  numbers,  and  it  is  required  to 
find  the  prime  factors  common  to  all  of  them  : 

Find  the  in'une  factors  of  carh,  and  then  select  thoae  factors 
ivhich  are  common  to  all  the  members. 

8.  What  are  the  puime  factors  common  to  150,  210  and  270  ? 

9.  What  are  the  prime  factors  common  to  42,  126,  and  168? 

10.  What  are  the  prime  factors  common  to  105,  315  and  525  ? 

11.  Wliat  are  the  prime  factors  common  to  84,  126  and  210? 

12.  What  are  the  prime  factors  common  to  168,  25.6,  410, 
and  820  ? 

13.  What  are  the  prime  factors  common  to  420,  630,  1050, 
and  2100  ? 

91.   DIVISIBILITY  OF  NUMBERS. 

1.  Two  is  the  only  even  number  which  is  prime. 

2.  Two  divides  every  even  number,  and  no  odd  number. 

3.  Three  divides  every  number  the  sum  of  Avhose  figures  is 
divisible  by  3. 

4.  Four  divides  every  number  when  the  two  right  hand 
figures  are  divisible  by  4. 

5.  Five  divides  every  number  which  ends  in  0  or  5. 

6.  Six  divides  every  even  number  that  is  divisible  by  3. 

7.  Ten  divides  every  number  ending  in  0. 

90.  How  do  you  find  the  prime  factors  common  to  several  numbers  ? 

91.  1.  How  many  even  numbers  arc  prims'!  2.  What  numliers  will  2 
divide  !  3  What  numbers  will  3  divide  1  4.  What  numbers  will  4  divide  1 
5.  What  numbers  will  ^  divide  1   6.  ^^'hat  numbers  will  6  divide  1   7.  What 


98  PKOPERTIES    OF    NUitBERS. 

8.  When  the  divisor  is  a  composite  number,  and  when  fhe 
iividend  and  partial  quotients  arc  successively  divisible  by  i/i 
factors,  the  division  will  be  exact  (Art.  57)- 

For,  dividing  by  the  factors  separately,  gives  the  same  quo 
tient  as  dividing  by  their  product  (Art.  70). 

9.  Any  number  u-hich  will  divide  one  factor  of  a  product  loil 
divide  the  product. 

Thus,  take  any  number,  as  30  =  5x6;  any  number  which 
will  divide  5  or  6  will  divide  30. 

10.  Any  number  which  will  exactly  divide  each  of  ta-o  num- 
bers will  divide  their  sum  :  and  any  number  which  will  divide 
their  sum  and  one  of  the  numbers,  toill  divide  the  other. 

For,  take  any  two  numbers,  an  9  and  12  ;  then, 

9  +  12  =  21. 
Now,  any  divisor   that   will   divide  two  of  these  numbers  will 
divide    the  other  ;  else,  we  should  have  a  whole  number  equal  to 
a  fraction,  which  is  impoi>sible. 

11.  Any  number  which  unll  exactly  divide  each  of  two  num- 
bers will  divide  their  difference  :  and  any  number  ivhich  will 
divide  their  diffirence  and  one  of  the  numbers,  will  divide  the 
other. 

For,  let  24  and  8  be  any  two  numbers ;  then, 

24-8  =  16. 
Now,  any  divisor  that  will  divide  two  of  these  numbers  will 
divide  the   other  ;  else,  we  shoidd  have  a  whole  number  equal  to 
u  fraction,  which  is  impossible. 

12.  Jf  there  is  a  remainder  after  division,  any  number  which 
wilt  exactly  divide  the  dividend  and  divisor  will  also  divide  the 
remainder. 


numbers  will  10  divide  1  8.  "When  will  llif  divi-or  exactly  divide  the  divi- 
dend !  9.  When  will  any  minihor  divide  a  p  rod  net  !  10.  \\  hen  u  ill  a 
number  divide  the  sum  of  two  numbers  !  When  will  it  divide  either  .)f 
them  separately  !  11.  When  will  a  number  exactly  divide  the  dilVereiico 
of  two  numbers  !  12.  If  a  nmnher  divides  the  dividend  and  divisor  whil 
kthcr  number  will  it   alwavs  divide'' 


GREATEST   COMMON    DIVISOR.  99 

For,  we  always  have 

Dividend  =  Divisor  x  Quotient  -|-  Eem. 
or  Dividend  —  Divisor  X  Quotient  =  Rem. ; 

hence,  by  principle  (11)  any  number  which  will  divide  the  divi- 
dend and  divisor  will  also  divide  the  remainder,  after  division. 

GREATEST  COMMON  DIVISOR. 

92.  A  Common  Divisor  of  two  or  more  numbers  is  any 
number  that  will  divide  each  of  them  without  a  remainder; 
hence,  it  is  merely  a  common  factor  of  the  numbers. 

93.  The  Greatest  Common  Divisor  of  two  or  more 
numbers  is  the  greatest  number  that  will  divide  each  of  them 
without  a  remainder;  hence,  it  is  their  greatest  coxanon  factor. 

For  example,  2  and  3  are  common  divisors  of  12  and  18; 
but  G  is  their  greatest  common  divisors,  since  there  is  no  num- 
ber greater  than  6  that  will  exactly  divide  both  of  them ;  hence, 
it  is  their  greatest  common  factor. 

Note. — Since  1  and  the  number  itself  will  divide  every  number, 
they  are  not  reckoned  among  the  common  divisors. 

Hence,  to  find  the  greatest  common  divisor  of  two  or  more 
numbers, 

I.  Resolve  each  number  into  its  2>i'ime  factors  : 

n.  Tlie  product  of  all  the  factors  common  to  each  result  will  be 
the  greatest  common  divisor. 

EXAMPLES. 

1.  "What  is  the  greatest  common  divisor  of  12  and  20? 

Analysis. — There  are  three  prime  fac- 
tors in  12 ;  viz.,  2,  2  and  3  :  there  are  three  operation. 
prime  factors  in  20;  viz.,  2,  2  and  5  :  the         12  =  2X2x3 
factors  2  and  2  are  common;  hence,  2x2  ==4         20  =  2x2x5 
is  the  greatest  common  divisor. 

92.  Wliat  is  a  common  divisor  of  two  or  more  numbers  ? 

93.  Wbat  is  the  sreatest  common  divisor  of  two  or  more  numbers  ?  Ho^v 

o 

4o  you  find  the  greatest  common  divisor  of  two  or  more  numbers  I 


100  GREATEST    COMMON    DIVISOR. 

2.  What  is  the  greatest  common  divisor  of  18  and  36. 

3.  What  is  the  greatest  common  divisor  of  12,  24  and  60 

4.  What  is  the  greatest  common  divisor  of  15,  50  and  40  ? 

5.  What  is  tlie  greatest  common  divisor  of  24,  18  and  144? 

6.  Wliat  is  the  greatest  common  divisor  of  50,  100  and  80  ? 

7.  What  is  the  greatest  common  divisor  of  56,  84  and  140  ? 

8.  What  is  tlie  greatest  common  divisor  of  84,  154  and  210  ? 

SECOND    METHOD, 

94.  When  the  numbers  are  large,  another  method  is  used  for 
finding  their  greatest  common  divisor. 

1.  Let  it  be  required  to  find  the  greatest  common  divisor  of 
the  numbers  216  and  408. 

Analysis. — The  greatest  common  divisor  opkration. 

cannot  be  greater  than  the  least  number  216.        216)408(1 
Now,  as  216  will  divide  itself,  let  us  see  if  it  216 

will  divide  408;  for  if  it  will,  it  is  the  great-  192)216(1 

est   common   divisor    sought.      Making   the  192 

division,  we  find  a  quotient  1  and  a  remain-  24)192(8 

der   192;  hence,   216    is    not  a  common  di-  192 

visor. 

The  greatest  common  divisor  of  216  and  408  will  divide  the 
remainder  192  (Art.  91-12);  and  if  192  will  exactly  divide  216,  it 
will  be  the  greatest  common  divisor.  Wc  find  that  192  is  contained 
in  216  once,  aud  a  remainder  24.  The  greatest  comtnon  divi.^^or  of 
192  and  216  will  divide  the  remainder  24;  and  if  24  will  exactly 
divide  192,  it  will  also  divide  216,  and  consequently  408  ;  now  24 
exactly  divides  192,  and  hence  is  the,  greatest  common  divisor  sought. 

Hence,  to  find  the  greatest  common  divisor, 

Divide  the  greater  number  hy  the  leas,  and  then,  divide  the 
preceding  divisor  by  the  remainder,  and  so  on,  till  nothing 
remains  :  the  last  divisor  will  be  the  greatest  coinmon  divisor. 

No  FES. — 1.  If  the  last  remainder  is  1,  the  numbers  haA^c  no  com 
mon  divisor ;  that  is,  they  are  prime  with  respect  to  each  other 
(Art.  87). 

94.  ^"^'l^at  is  tlic  rule  when  the  numbers  are  large  ? 


GREATEST    COMMON    DIVISOK.  101 

2.  If,  in  the  course  of  tne  opcralion,  any  one  of  the  remainders  is 
a  prime  number^  and  will  not  exactly  divide  the  ■preceding  divisor,  it 
is  certain  that  no  common  divisor  exists,  aud  it  is  unnecessary  to 
divide  further. 

EXAMPLES. 

1.  "What  is  the  greatest  common  divisor  of  3328  and  4592  ? 

2.  What  is  the  greatest  common  divisor  of  2205  and  4501  ? 

3.  What  is  the  greatest  number  that  -will  divide  1G082  and 
25740  ? 

4.  What  is  the  greatest  number  that  will  divide  620,  1116 
and  1488  ? 

5.  What  is  the  greatest  common  divisor  of  5270,  5952,  5394 
and  3038  ? 

6.  What  is  the  greatest  common  divisor  of  4617,  7695,  6642 
and  8424  ? 

7.  A  farmer  has  315  bushels  of  corn,  and  810  bushels  of 
wheat ;  he  wishes  to  draw  the  corn  and  wdieat  to  market 
separately  in  the  fewest  number  of  equal  loads  :  how  many- 
bushels  must  he  draw  at  a  load  ? 

8.  The  Illinois  Central  Railroad  Company  have  15750  acres 
of  land  in  one  location,  and  21725  acres  in  another.  They 
wish  to  divide  the  whole  into  lots  of  equal  extent,  containing 
the  greatest  number  of  acres  that  will  give  an  exact  division : 
how  many  acres  will  there  be  in  each  lot  ? 

9.  A  man  has  a  corner  lot  of  land  1044  feet  long,  and  744 
feet  wide.  The  adjacent  sides  are  bounded  by  the  highway 
and  he  wishes  to  build  a  boai'd  fence  with  the  fewest  panels  of 
equal  length  :  what  must  be  the  length  of  the  panels  ? 

10.  A  farmer  has  231  bushels  of  barley,  369  bushels  of  oats, 
and  393  bushels  of  wheat,'  all  of  which  he  wishes  to  put  into 
the  smallest  number  of  bags  of  equal  size,  without  mixing : 
how  many  bushels  must  each  bag  contain  ? 

11.  Three  persons.  A,  B,  and  C,  agree  to  purchase  a  lot 
of  63  cows  at  the  same  price  per  head,  provided  each  man  can 
thus  invest  his  whole  money.  A  has  $286,  B  .^462,  and  C  8638  ; 
how  many  cows  could  each  man  purchase  ? 


102  LEAST    COMMON    MULTIPLE. 

LEAST    COMMON    MULTIPLE. 

95.  A  Multiple  of  a  number  is  any  product  in  -which  th« 
number  enters  as  a  factor ;  hence,  a  muhiple  of  any  number  is 
exactly  divisible  by  the  number. 

96.  A  Common  Multiple  of  two  or  more  numbers  is  any 
number  which  each  -will  divide  -without  a  remainder. 

97.  The  Least  Common  Multiple  of  two  or  more  num- 
bers is  the  least  number  -svhich  they  will -separately  divide  with 
out  a  remainder. 

Notes. — 1.  Since  the  least  common  multiple  is  exactly  divisible 
by  a  divisor,  it  can  be  resolved  into  two  factors,  one  of  which  is  the 
divisor  and  tlie  other  the  quotient. 

2.  !f  the  divisor  be  resolved  into  its  prime  factors,  the  equal  factoi 
of  the  least  common  multiple  maybe  resolved  into  the  same  factors 
hence,  the  least  common  multiple  will  contain  every  prime  factor  of  it. 
divisor. 

3.  The  question  of  finding  the  least  common  multiple  of  several 
numbers,  is  therefore  reduced  to  finding  a  number  which  shall  con- 
tain all  their  prime  factors  and  none  others. 

1.  What  is  the  least  common  multiple  of  6,  12  and  18  ? 

A.NALTSIS. — Havmg  placed  the  given  num- 
bers in  a  line,  if  we  divide  by  2,  we  find  the  operation. 
quotients  3,  6  and  9  ;  hence,  2  is  a  prime  fac-        2)6  .  .   12   .  .   18 
tor  of  all  the  numbers.      Dividing  by  3,  -we        3)3  .   .     6  .  .     9 
find  that  3  is  a  prime  factor  of  the  quoti-             1   .  .     2  .  .     3 
ents  3,  6,  and  9  ;  and  hence,   the   quotients         2X3X2x3  =  36 
2   and  3  are  prime   factors  of   12   and   18; 

hence,  the  prime  factors  of  all  the  numbers  are  2,  3,  2  and  3,  and 
their  product  36  is  the  least  common  multiple. 

98.  Therefore,  to  find  the  least  common  multiple  of  several 
numbers  : 

95.  Wliat  is  a  multiple  of  a  number  \ 

96.  What  is  a  common  multiple  of  two  or  more  numbers  1 

97.  What  is  the  least  common  multiple  of  two  or  more  numbers  ? 
9S.   How  do  you  find  the  least  common  multiple  ol  several  numbers' 


COMMON    MULTIPLIi.  103 

I.  Place  the  numbers  on  the  same  line,  and  divide  by  any  prime 
number  that  tuill  exactly  divide  tiuo  or  more  of  them,  and  set 
doivn  in  a  line  below  the  quotients  and  the  undivided  members. 

II.  Then  divide  as  before,  until  there  is  no  prime  number 
greater  than  1  that  will  exactly  divide  any  two  of  the  numbers. 

III.  Then  multiply  together  the  divisors  and  the  numbers  of 
the  lower  line,  and  their  product  tvill  be  the  least  common 
multiple. 

Note. — If  the  numbers  have  no  common  prime  factor,  their  pro- 
duct will  be  their  least  common  midtiple. 

EXAMPLES. 

1.  What  is  the  least  common  multiple  of  4,  9,  10,  15,  18, 
20,  21  ? 

2.  What  is  the  least  common  multiple  of  8,  9,  10,  12,  25, 
32,  75,  80  ? 

3.  What  is  the  least  common  multiple  of  1,  2,  3,  4,  5,  6, 
7,9? 

4.  What  is  the  least  common  multiple  of  9,  16,  42,  63,  21 
14,  72  ? 

5.  What  is  the  least  common  multiple  of  7,  15,  21,  28,  35, 
100,  125  ? 

6.  What  is  the  least  common  multiple  of  15,  16,  18,  20,  24, 
25,  27,  30  ? 

7.  What  is  the  least  common  multiple  of  9,  18,  27,  36,  45^ 
54? 

8.  What  is  the  least  common  multiple  of  4,  10,  14,  15,  21  ? 

9.  What  is  the  least  common  multiple  of  7,  14,  16,  21,  24? 

10.  What  is  the  least  common  multiple  of  49,  14,  84,  108, 
98? 

11.  A  can  dig  9  rods  of  ditch  in  a  day;  B  12  rods  in  a  day; 
and  C  1 6  rods  in  a  day :  what  is  the  smallest  number  of  rods 
that  would  afford  exact  days  of  labor  to  each,  working  alone  ? 
In  what  time  would  each  do  the  whole  work  ? 

12.  A  blacksmith  employed  4  classes  of  workmen,  at  $15, 
$16,  121  and  $24  per  month,  for  eacli  man  respectively,  paying 
to  each  class  the  same  amount  of  wages.     Required  the  least 


104  CANCELLATION. 

amount  that  will   j^ay  either  class  for  1  month  ;  also,  the  num- 
ber of  men  in  each  class  ? 

13.  A  farmer  has  a  number  of  ba"js  containinsr  2  bushels 
each  ;  of  barrels,  containing  3  bushels  each  ;  of  boxes,  containing 
7  bushels  each ;  and  of  hogsheads,  containing  15  bushels  each : 
what  is  the  smallest  quantity  of  wheat  that  would  fill  each  an 
exact  number  of  times,  and  hoio  many  times  would  that  quan- 
tity fill  each  ? 

14.  Four  persons  start  from  the  same  point  to  travel  round 
a  circuit  of  300  miles  in  circumfei'cnce.  A  goes  15  miles  a 
day,  B  20  miles,  C  25  miles,  and  D  30  miles  a  day.  How 
many  days  must  they  travel  before  they  will  all  come  together 
again  at  the  same  point,  and  how  many  times  will  each  have 
gone  round  ? 

Note. — First  find  the  number  of  days  that  it  will  take  each  to 
travel  round  the  circuit. 

CANCELLATION. 

99.  Cancellation  is  a  method  of  shortening  Arithmetical 
operations  by  omitting  or  cancelling  common  factors. 

1.  Divide  36  by  18.     First,  36  =  9  x  4 ;  and  18  =  9  x  2 


2. 


Analysis. — Thirty-six  divided  by  18  is  operation. 

equal  to  9  X  4  divided  by  9  x  2  :  by  can-  36         0  x  4  _ 

celling,  or  striking  out  the  9's,  we  have  18  ~    0X2 
4  divided  by  2,  which  is  equal  to  2. 

Note. — The  figures  cancelled  arc  slightly  crossed. 

The  operations,  in  cancellation,  depend  on  two  principles  : 

1 .  JVie  cancelling  of  a  factor,  in  any  number,  is  equivalent  to 
dividing  the  number  by  that  factor 

2.  If  the  dividend  and  divisor  he  both  divided  by  the  same 
number,  the  quotient  will  not  be  changed. 


99.  What  is  cancellation  1     On  what  principles  do  the  operations  of 
cancellation  depend  ? 


CANCELLATION.  105 

PRINCIPLES    AND    EXAMPLES. 

1.  Divide  5G  by  32. 

Analysis. — Resolve  the  dividend   and     '  operation. 

divisor  into  factors,  and  tlien  cancel  those  56       S  x  7       7 


which  are  common.  32      ;8  x  4       4 

2.  In  72  times  25,  how  many  times  45  ? 

Analysis. — We  see  that  9  is  a  factor  of  72  and  45.     Divide  by  9, 
and  write  the  q^uotient  8  over  72,  and  the 
quotient  5  below  45.     Again,  5  is  a  fac-  operation. 

tor  of  25  and  5.     Divide   25  by  5,   and  8         5 

write   the  quotient  5  over  25.     Dividing  "^^  ^  ^-■^  _  .„ 

5  by  5,  reduces   the  divisor  to   1,  which  A? 

need  not  be  set  down  :  hencC;  the  true 
quotient  is  40. 

Note. — The  operation  may  be  performed  in  another  way,  by  wri- 
ting the  divisor  on  the  left  of  a  vertical 

line,   and  the  dividend  on  the  right:    in  operation. 

which  case,  the  quotients,  in  cancelling,  ^ 

are  written  above,  and  at  the  side  of  the 
numbers,  as  5,  8  and  5.     If  we  conceive 


5 

the  horizontal  line,  first  used,  to  be  turn-  a        .^ 

ed  up  from  left  to  right,    the    dividend, 

which  was   above  the  line,  will  fall  at  the  right,  and  the  divisor, 

which  was  below  it,  at  the  left. 

100.  Hence,  to  perform  the  operations  of  cancellation  : 

I.  Resolve  the  dividend  and  divisor  into  such  factors  as  shall 
give  all  the  factors  common  to  both. 

II.  Cancel  the  common  factors  and  then  divide  the  product  of 
the  remaining  factors  of  the  dividend  hy  the  product  of  the  re- 
maining factors  of  the  divisor. 

Notes. — 1.  Since  every  factor  is  cancelled  by  division^  the  quotient 
1  always  takes  the  place  of  the  cancelled  factor,  but  is  omitted  when 
it  is  a  multiplier  of  other  factors. 

2  If  one  of  the  numbers  contains  a  factor  equal  to  the  product  of 
two  or  more  factors  of  the  other,  all  such  factors  may  be  cancelled. 

100.  How  do  you  perform  the  operations  of  cancellation  T 


106  OA]S-CELLATION. 

3.  If  the  product  of  two  or  more  factors  of  the  dividend  is  equal  to 
the  product  of  two  or  more  factors  of  the  divisor,  such  factors  may  be 
cancelled. 

4.  It  is  generally  more  convenient  to  set  the  dividend  on  the  right 
of  a  vertical  line  and  the  divisor  on  the  left. 

EXAMPLES. 

1.  What  is  the  quotient  of  2x4x8x13x7x16  divid- 
ed bj  26  X  14  X  8  ? 

2.  What  is  the  quotient  of  42  X  3  X  25  x  12  divided  hj 
28  X  4  X  15  X  6? 

3.  What  is  the  quotient  of  125  x  60  X  24  X  42  divided  by 
25  X  120  X  36  X  5? 

4.  How  many  times  isllx39x7x2  contained  in  44  x 
18  X  26  X  14? 

5.  What  is  the  quotient  of  8  times  240  multiplied  by  5  times 
114,  divided  by  24  time.s  57  multiplied  by  6  times  15  ? 

6.  What  is  the  value  of  (22  +  8  +  16)  X  (18  +  10  +  21) 
divided  by  (9  +  5  +  7)  x  (15  +  8)  ? 

7.  Divide (140  +  86  -  34)  x  (107  -  19)  by  (237  -  141) 
X  (17  +  20  -  15)  ? 

8.  Divide  [12  X  5  -  2  X  9]  x  (42  +  30)  by  (5  x  8) 
X  (2  X  9)  X  (10  +  17)  ? 

9.  What  is  the  quotient  of  240  x  441  X  16  divided  by 
175  X  56  X  27? 

10.  What  is  the  quotient  of  64  times  840  multiplied  by 
9  times  124,  divided  by  32  times  560  multiplied  by  4  times  31  ? 

11.  How  many  dozens  of  eggs,  worth  14  cents  a  dozen,  must 
be  given  for  18  pounds  of  sugar,  woi'th  7  cents  a  pound  ? 

12.  A  dairyman  sold  5  cheeses,  each  weigliing  40  pounds,  at 
9  cents  a  pound :  how  many  pounds  of  tea,  worth  50  cents  a 
pound,  must  he  receive  for  the  cheeses  ? 

13.  Bought  12  yards  of  cloth  at  Sl,84  a  yard,  and  paid  for 
it  in  potatoes  at  48  cents  a  bushel :  how  many  bushels  of  pota- 
toes will  pay  for  the  cloth  ? 

14.  How  many  firkins  of  butter,  each  containing  56  pounds, 


CANCELLATION.  107 

at  25   cents   a  pound,  will  pay   for   4  barrels  of  sugar,  each 
weighing  175  pounds,  at  8  cents  a  pound  ? 

15.  A  man  bought  10  cords  of  wood,  at  20  shillings  a  cord, 
and  paid  in  labor  at  12  shillings  a  day  :  how  many  days  must 
he  labor  ? 

16.  How  many  pieces  of  cloth,  each  containing  36  yards,  av 
$3,50  a  yard,  must  be  given  for  96  barrels  of  flour,  at  $10,50  a 
barrel ? 

17.  A  farmer  exchanged  492  bushels  of  wheat,  worth  §1,84 
a  bushel,  for  an  equal  number  of  bushels  of  barley,  at  87  cents 
a  bushel,  of  corn  at  GO  cents  a  bushel,  and  of  oats  at  45  cents  a 
bushel :  how  many  bushels  of  each  did  he  receive  ? 

IS.  How  many  barrels  of  flour,  worth  $7  a  barrel,  must  be 
given  for  250  bushels  of  oats,  at  42  cents  a  bushel  ? 

19.  If  48  acres  of  land  produce  2484  bushels  of  corn,  how 
many  bushels  will  120  acres  produce? 

20.  A  man  works  12  days  at  9  shillings  a  day,  and  receives 
in  pay  wheat  at  two  dollars  a  bushel :  how  many  bushels  did 
he  receive  ? 

21.  A  grocer  sold  6  hams,  each  weighing  14  pounds,  at  10 
cents  a  pound,  and  took  his  pay  in  apples  at  48  cents  a  bushel : 
how  many  bushels  of  apples  did  he  receive? 

22.  How  long  will  it  take  a  man,  travelling  36  miles  a  day, 
to  go  the  same  distance  that  another  man  has  travelled  in  15 
days  at  the  rate  of  27  miles  a  day  ? 

23.  A  man  took  4  loads  of  apples  to  market,  each  load  con- 
tainins:  12  barrels,  and  each  barrel  3  bushels.  He  sells  them 
at  45  cents  a  bushel,  and  receives  in  payment,  a  number  of 
boxes  of  tea,  each  box  containing  20  pounds,  Avorth  72  cents  a 
pound  :  how'  many  boxes  of  tea  did  he  receive  ? 


108  COMMON    FRACTIONS. 


COMMON  FRACTIONS. 

101.  The  unit  1  denotes  an  entire  thing,  as  1  apple,  1  chair, 
1  pound  of  tea. 

If  the  unit  1  be  divided  into  two  equal  parts,  each  part  is 
called  one-half. 

If  the  unit  1  be  divided  into  three  equal  parts,  each  part  is 
called  one-third. 

If  the  unit  I  be  divided  into  four  equal  parts,  each  part  is 
called  one-fourth. 

If  the  unit  1  be  divided  into  twelve  equal  parts,  each  part  is 
called  one-twelfth  ;  and  if  it  be  divided  into  any  number  of  equal 
parts,  we  have  a  like  expression  for  each  part. 

The  parts  are  thus  written : 


1  is  read,  one-half. 
1     -     -      one-third. 
J-     -     -      one-fourth, 

4 

i     -     -      one-liflh. 
i     -     -      one-sixth. 


1    is  read,  one-seventh. 
•|-      -     -      one-eighth. 

one-tenth. 

one-fifteenth. 

one-fiftieth. 


1  0 


15 

JL 

50 


The  ^,  is  an  entire  half ;  the  1,  an  entire  third ;  the  1,  an 
entire  fourth  ;  and  the  same  for  each  of  the  other  equal  parts ; 
hence,  each  equal  part  is  an  entire  thing,  and  is  called  a  frac- 
tional unit. 

The  unit,  or  whole  thing  which  is  divided,  is  called  the  unit 
of  the  fraction. 

NoTK. — In  every  fraction  let  the  pupil  distinguish  carefully,  be- 
tween the  unit  of  the  fraction  and  the  fractional  unit.  The  first  is 
the  whole  thing  from  which  the  fraction  is  derived;  the  second,  orie 
of  the  equal  parts  into  which  that  thing  is  divided. 

101.  Wha»  is  a  unit  1  What  is  each  part  called  when  the  unit  1  is  divid- 
ed into  two  equal  parts  !     When  it  is  divided  into  3]     Into  il     Into  51 

Into  lai 


COMMON    FRACTIONS.  109 

102.  Each  fractional  unit  may,  like  the  unit  1,  become  the 
base  of  a  collection  :  thus,  suppose  it  were  required  to  express 
2  of  each  of  the  fractional  units  :  we  should  then  write 


2 
2 

which  is  read 

2  halves  =  i  x  2. 

2 
3 

_     .     .     . 

2  thirds   =  ^  x  2. 

2 
4 

_     -     -     _ 

2  fourths  =  1x2. 

* 

-     -     .     _ 

2  fifths     =  i  X  2. 

&;c.,           &;c., 

&c.,           &c. 

If  it  were  required  to  express  3  of  each  of  the  fractional 
units,  we  should  write 

|-  which  is  read  3  halves  —1x3. 
f  -  -  -  -  3  thirds  =1x3. 
f  -  -  -  -  3  fourths  =  1x3. 
3  .  .  .  .  3  fifths  =1x3. 
&c.,       &c.,       &c.,       &c. ;  hence, 

A  Fraction  is  one  of  the  equal  parts  of  a  unit,  or  a  collec- 
tion of  such  equal  parts. 

Eractions  are  expressed  by  two  numbers,  one  written  above 
the  other,  with  a  line  between  them.  The  lower  number  is 
called  the  denominator,  and  the  upper  number  the  numerator. 

The  denominator  denotes  the  number  of  equal  parts  into 
which  the  unit  is  divided  ;  and  hence,  determines  the  value  of 
the  fractional  unit.  Thus,  if  the  denominator  is  2,  the  fractional 
unit  is  one-half;  if  it  is  3,  the  fractional  unit  is  one-third  ;  if  it 
is  4,  the  fractional  unit  is  one-fourth,  &cc.,  Sec. 

The  numerator  denotes  the  number  of  fractional  units  taken. 
Thus,  f  denotes  that  the  fractional  unit  is  i,  and  that  3  such 
units  are  taken;  and  similarly  for  other  fractions. 

How  may  the  one-half  be  regarded  1  The  one-third  ?  The  one  fourth  ? 
What  is  each  part  called  !  "What  is  the  unit  of  a  fraction  1  What  is  a 
fractional  unit  !     How  do  you  distinguish  between  the  one  and  the  other  ! 

102.  May  a  fractional  unit  become  the  base  of  a  collection  1  What  is  a 
fraction  ?  How  are  fractions  expressed  ?  What  is  the  lower  number 
called  ?  What  is  the  uppemumber  called  \  What  does  the  denominator 
denote  7     What  does  trhe  numerator  denote  !     In  the  fraction  .'}  fifths,  what 


110  COMMON    FRACTIONS. 

In  the  fraction  f ,  the  base  of  the  collection  of  fractional  units 
is  ^,  but  this  is  not  the  primary  base.  For,  ^  is  one-Jrfth  of  the 
unit  1  ;  hence,  the  primary  base  of  every  fraction  is  the  u?iit  1. 

103.  If  we  suppose  a  second  unit  to  be  divided  into  the 
eame  number  of  equal  parts,  such  parts  may  be  expi'essed  in 
the  same  collection  with  the  parts  of  the  first :  thus, 

I     is  read  3  halves. 

•^     -     -     -  7  fourths. 

i_6     .     .     .  16  fifths. 

i_8     .     .     .  18  sixths. 

^-f     -     -     -  25  sevenths. 

104.  A  %\hole  number  may  be  expressed  fractionally  by 
writing  1  below  it  for  a  denominator.     Thus, 

3  may  be  written  y  and  is  read,  3  ones. 

5  --.-       A       ---5  ones. 

6  ----       |.       -..      6  ones. 
8  ----       ^       _._      8  ones. 

But  3  ones  are  equal  to  3,  5  ones  to  5,  6  ones  to  6,  and 
8  ones  to  8  ;  hence,  the  value  of  a  number  is  not  changed  by 
placing  1  under  it  for  a  denominator. 

105.  If  the  numerator  of  a  fraction  be  divided  by  its  denomi- 
nator, the  integral  part  of  the  quotient  will  express  the  number 
of  entire  units  used  in  forming  the  fraction ;  and  the  remainder 
will  show  how  many  fractional  units  are  over.  Thus,  y  are 
equal  to  3  and  2  thirds,  and  is  written  y  =  3|- :  hence, 

A  fraction  has  the  same  form  as  an  unexecuted  division. 

is  the  fractional  base  1     What  is  the  primary  base  1     What  is  the  primary 
base  of  every  fraction  1 

103.  If  a  second  unit  be  divided  into  the  same  number  of  equal  parts, 
may  the  parts  be  expressed  with  those  of  the  first  I  How  many  unita 
have  been  divided  to  obtain  G  thirds  1     To  obtain  9  halves  1     12  fourths  1 

104.  How  may  a  whole  number  bo  expressed  fractionally]  Does  this 
chanrre  the  value  of  the  number  ! 

105.  If  the  numerator  be  divided  by  the  denominator,  what  does  the 
quotient  show  1  What  does  the  remainder  show  1  What  form  has  a  frai> 
lion  1     Wluit  arc  the  seven  i)rinciplc8  which  foiliiw  ? 


COMMON    FRACTIONS.  Ill 

From  what  has  been  said,  we  conclude  that, 

1st.  A  fraction  is  one  of  the  equal  parts  of  a  unit,  or  a  col- 
lectioji  of  such  equal  parts  : 

2d.  Tlie  denominator  shous  into  hoxo  many  equal  parts  the  unit 
is  divided,  and  hence  indicates  the  vcdue  of  the  fractiomd  unit  : 

3d.   27ie  numerator  shows  how  inany  fractional  units  are  taken  . 

4th.  'llie  value  of  every  fraction  is  equal  to  the  quotient  arising 
from  dividing  the  numerator  hy  the  denominator : 

5th.  When  the  numerator  is  less  than  the  denominator,  the  value 
of  the  fraction  is  less  than  1  : 

6th.  When  the  numerator  is  equal  to  the  denominator,  the  value 
of  the  fraction  is  equal  to  1  : 

7th.  When  the  numerator  is  greater  than  the  denominator,  the 
value  of  the  fraction  is  greater  than  1. 

EXA3IPLES    IN    WRITING    AND    READING    FRACTIONS. 

1.  Read  the  following  fractions  : 

S  7  5         _6  2J         ]_6        j_a_ 

9'        12'        3'        15J         9   '  7   '        104' 

What  is  the  unit  of  the  fraction,  and  what  the  fractional  unit  in 
each  example  ?     How  many  fractional  units  are  taken  in  each  ? 

2.  Write  15  of  the  19  equal  parts  of  1.  Also,  37  of  the  49 
equal  parts  of  1. 

3.  If  the  unit  of  the  fraction  is  1,  and  the  fractional  unit 
one-fortieth,  express  27  fractional  units.  Also,  95.  Also,  106. 
Also,  87.     Also,  41. 

4.  If  the  unit  of  the  fraction  is  1,  and  the  fractional  unit 
one  68th,  express  45  fractional  units.  Also,  56.  Also,  85. 
Also,  95.     Also,  37, 

5.  If  the  unit  of  the  fraction  is  1,  and  the  fractional  unit  one 
90th,  express  9  fractional  units.  Also,  87.  Also,  75.  Also,  65. 
Also,  85.     Also,  90.     Also,  100. 

DEFINITIONS. 

106.  A  Proper  Fraction  is  one  whose  numerator  is  less 
than  the  denominator. 

100.  What  is  a  proper  fraction  ?     Give  examples. 


112  COMMOX    FK ACTIONS. 

The  following  are  proper  fractions  : 

1     JL     1      3.      3^     S      JL      8.      5 

2'     3'     4'     4'     7'     8'     10'     9'     6' 

107.  Ax  Improper  Fraction  is  one  whose  numerator  is 
equal  to,  or  exceeds  the  denominator. 

Note. — Such  a  fraction  is  called  improper  because  its  value  equals 
or  exceeds  1. 

The  following  are  improper  fractions : 

3      A      6.      8^     1     12.     JLt     ±9_ 

2'    3'    5'     7'    8'      6  '      7  '      7   * 

108.  A  Simple  Fraction  is  one  whose  numerator  and  de- 
nominator are  both  whole  numbers. 

Note. — A  simple  fraction  may  be  either  proper  or  improper. 
The  following  are  simple  fractions  : 

J.3^5_8.98..6X 
4'     2'     6'     7'     2'    3'    3'     5* 

109.  A  Compound  Fraction  is  a  fraction  of  a  fraction,  or 
eeveral  fractions  connected  by  the  word  of. 

The  following  are  compound  fractions  : 

1  of  1      1  of  1  of  1      i  of  3      J-  of  1  of  4 

2^4'        3    "^    2  3'         6  '         7    Ul    g    Ui   "i. 

110.  A  Mixed  Number  is  made  up  of  a  whole  number  and 
a  fraction. 

The  following  are  mixed  numbers  : 

31,         41,         6f,         5f,         Gf,         31. 

111.  A  Complex  Fraction  is  one  whose  numerator  or  de- 
nominator is  fractional ;  or,  in  which  both  are  fractional. 

The  following  are  complex  fractions  : 

{})  J_  (t)  ^ 

5'  191'  (!')  691- 

107.  Wliat  is  an  improper  fraction?     Wliy  improper  1     Give  examples 

108.  Wliat  is  a  simple  fraction  1     Give  examples.     May  it  be  proper  or 
improper  1 

109.  M'liat  is  a  compound  fraction  1     Give  examples, 

110.  ^\■hat  is  a  mixed  numberl     Give  examples, 

111.  What  is  a  complex  fraction  1     Give  examples. 


COMMON    FJRACTIONS.  113 

112.  The  numerator  and  denominator  of  a  fraction,  taken 
together,  are  called  the  terms  of  the  fraction :  hence,  every 
fraction  has  two  terms. 

FUNDAMENTAL    PROPOSITIONS. 

113.  By  multiplying  the  unit  1,  we  form  all  the  whole 
numbers, 

2,     3,     4,     5,     G,     7,     8,     9,     10,     &c.; 

and  by  dividing  the  unit  1  by  these  numbers  we  form  all  the 
fractional  units, 

2'        3'        4'        5'         6'         7'         8'         9'        TO'        "^^* 

Now,  since  in  1  unit  there  are  2  halves,  3  thirds,  4  fourths, 
5  fifths,  6  sixths,  &;c.,  it  follows  that  the  fractional  unit  becomes 
less  as  the  denominators  are  increased:  hence, 

Any  fractional  imit  is  such  apaj-i  ofl,  as  1  is  of  the  denominator. 

Thus,  ^  is  one-half  of  1,  since  1  is  one-half  of  the  denomi- 
nator 2  ;  i  is  one-third  of  1,  since  1  is  one-third  of  3  ;  i  is  one 
fourth  of  1  ;  |-,  one-fifth  of  1,  &c.  &c. 

114.  Let  it  be  required  to  multiply  ^  by  4. 

Analysis. — In  f  there  arc  3  fractional  units,  operation. 

each  of  which  is  i,  and  these  are  to  be  taken     -1x4=:  ^^  =  J^. 
4  times.     But  .3  things  taken  4  times,  gives  12 

things  of  the  same  kind ;  that  is,  12  eighths  ;  hence,  the  product  is  4 
times  as  great  as  the  multiplicand  :  therefore,  we  have 

Proposition  I. — If  the  numerator  of  a  fraction  he  multiijlicd 
by  any  number^  the  fraction  will  he  muIti'pUed  as  many  times  as 
there  are  units  in  that  number. 


112.  How  many  terms  has  every  fraction  1     "What  are  they  1 

113.  How  may  all  the  whole  numbers  be  formed  ?  How  may  the  frac- 
tional units  be  formed  ?  What  part  of  one,  is  one-half]  What  part  of  1 
is  any  fractional  unit  ] 

114.  What  is  proved  in  proposition  11 

6 


114 


PRINCIPLES    OP 


EXAMPLES. 


1.  Multiply  f  by  6,  by  7. 

2.  Multiply  I  by  4,  by  9. 

3.  Multiply  g?,- by  11,  by  12. 

4.  Multiply^^by  12,  by  14, 
115.  Let  it  be  required  to  multiply  -^  by  4. 
Analysis. — In  ^  there  are  5  fractional 
lits,  each  of  which  is  -^^-     ^^  ''^'®  '^'"^"ide  operation. 

the  denominator  by  4,  the  quotient  is  3,  and      -^  X  4  r=  -pr^ 


5.  Multiply  fj-  by  3,  by  4. 

G.  Multiply  If  by  7,  by  9. 

7.  Multiply  A|.  by  5,  by  10. 

8.  Multiply -1^- by  3,  by  11. 


the  fractional  unit  becomes  ^,  which  is  4 
times  as  great  as  -^j  because,  if  ^  be  divided  into  4  equal  parts  each 
part  will  be  j\j.  If  wc  take  this  fractional  unit  5  times,  the  result 
^  will  be  4  times  as  great  as  ^ ;  therefore,  we  have 

Proposition  II. — If  the  denominator  of  a  fraction  be  divided 
by  any  number^  the  value  of  the  fraction  will  be  increased  as 
many  times  as  there  are  units  in  that  number. 


EXAMPLES. 


1.  Multiply  I  by  2,  by  4. 


2.  Multiply  1^  by  8,  by  4,  2. 

3.  Multiply  ^\  by  2,  3,  4, 
6,8. 

4.  Multiply  ^  by  6,  by  5, 
10,  15. 

5.  Multiply  -}|  by  2,  3,  4,  G, 
8,  12,  16,  and  24. 

116.   Let  it  be  required  to  divide  -^j  by  3 


G.  Multiply  fo  by  2,  4,  5, 
10,  20. 

7.  Multiply   3^3-   by   7,    and 
by  5. 

8.  Multiply  ^y  by  21,  6,  7, 
3,  and  2. 

9.  Multiply  i;i-  by  2,  3,  4,  % 
9,  and  12. 


Analysis. — In  -,"1.  there  are  9  fractional 


1 1 


OPERATION. 


9  ^3  _    8 


units,  each  of  whic'.i  is  -jlj,  and  these  are  to      ^^  -7-  3 

be  divided  by  3.     But  9  things,  divided  by 

3,  gives  3  things  of  the  same  kind  lor  a  quotient  :  hence,  the  quotient 

is  3  elevenths,  a  number  Avhich  is  one-third  of  ^^  ;  hence,  we  have 

Proposition  III. — If  the  nuuxrator  of  a  fraction  be  divided 
Inj  any  number,  the  fraction  will  be  diminished  as  many  times  as 
there  are  units  in  that  number. 


11. "j.   Wiiat  is  proved  in  projiosition  11  ! 
116.   What  in  piovcd  in  proposition  III ! 


COMMON   FKACTIONS. 


115 


EXAMPLES. 


1.  DivideJf  by2,4,8, 16. 

2.  Divide   jf    by  2,  7   and 


14. 


3.  Divide  ^  by  2,  5,  4  and 


10. 


4.  Divide  fl   by  5,  G,  10, 
15  and  20. 


5.  Divide  l|  by  2,  3,  6,  and 
G.  Divide  If  by  3,  6,  8,  and 


12. 


7.  Divide  f |  by  3,  9  and  27 

8.  Divide  fA  by  G,   9,  27 
and  54. 

117.   Let  it  be  required  to  divide  -^  by  3. 

Analysis. — In  -^,  there  are  9  fractional  operation. 

units,  each   of  which  is  ^V-     Now,  if  we      J--:_3  =  -? —  =  A. 

;  11  '  11-  11X833 

multiply  the  denominator  by  3,  it  becomes 

33.  and  the  fraciioual  unit  becomes -^j  which  is  one-third  part  of 
^.  If,  then,  we  take  this  fraciioiial  unit  9  times,  the  result  -^  is 
just  one-tliird  part  of  ^j  :  hence,  we  have  divided  the  fraction  y\ 
by  3  :  therefore,  we  have, 

Proposition  IV. — Jf  the  denominator  of  a  fraction  he  mul- 
tiplied hy  any  niimhrr,  the  fraction  will  be  diminished  as  many 
times  as  there  are  units  in  that  number. 


EXAMPLES. 


1.  Divide  f  by  G,  7  and  8. 

2.  Divide  |  by  5,  4  and  9. 

3.  Divide  If  by  3,  4  and  12. 

4.  Divideff  byG,  8andll. 


5.  Divide  |f  by  7,  5  and  3. 

G.  Divide  \\  by  7,  8  and  6. 

7.  Dividers  by  3,  7  and  11. 

8.  Divide  -fi  by  8,  4  and  10. 

118.   Let  it  be  required  to  multiply  both  terms  of  the  frac- 
tion 1^  by  4. 

Analysis. — In  f .  the  fractional  unit  is  -^,  and 
it  is  taken  3  times.   By  multiplying  the  denomi- 
nator by  4.  the  fractional  unit  becomes  -^.  the 
value  of  which  is  is  one-fourth  of  \.   By  multiplying  the  numerator 
by  4.  we  increase  the  number  of  fraclional  units  taken.  4  times  ;   that 


operation. 

3X4    12^ 

5X4    ~   '2  0 


117.  If  the  denominator  of  a  fraction  be  multiplied  by  any  number,  how 
will  the  value  of  the  fraction  be  effected  ? 

118.  If  both  terms  of  a  fraction  be  multiplied  by  any  number,  how  will 
the  value  of  the  fraction  be  effected  ^ 


116  PRINCIPLES   OF 

is,  we  increase  the  number  of  parts  taken  just  as  many  times  as  we 
decrease  the  value  of  the  fractional  unit  ;  hence  the  value  of  the  trac- 
tion is  not  changed  :  therefore,  we  have 

Pkoe^sition  V. — //■  both  terms  of  a  fraction  he  multl^^lied 
by  the  same  number,  the  value  of  the  fraction  will  not  be 
changed. 

EXAMPLES. 

1.  Multiply  both  terms  of  the  fraction  |-  by  4,  by  6,  and  by  5. 

2.  Multiply  both  terms  of  -fj  by  5,  by  8,  by  9,  and  11. 

3.  Multiply  both  terms  of  if  by  7,  by  8,  and  9. 

4.  Multiply  both  terms  i^-  by  5,  8,  6,  and  12. 

5.  Multiply  both  terms  of  |-|  by  2,  3,  4,  and  5. 

119.  Let  it  be  required  to  divide  the  numerator  and  denomi. 
nator  of  -^  by  3. 

Analysis. — In  ^^,  the  fractional   unit  is  ■^,         operation. 
and  it  is  taken  6  times.  By  dividing  the  dcnonii-  6  h-3^2 

nator   by  3.  the  fractional  unit  becomes  \,  the  15^3     5 

value  of  which  is  3  times  as  great  as  ■^^.     By  ; 

dividing  the  numerator  by  3,  we  diminish  the  number  of  fractional 
luiits  taken  3  times  ;  that  is.  we  diniinish  the  number  of  parts  taken 
just  as  many  ti77ies  as  we  increase  the  value  of  the  fractional  tinit : 
hence,  the  value  of  the  fraction  is  not  changed  ;  therefore,  we  have 

Proposition  VI. — If  both  terms  of  a  fraction  be  divided  by 
the  same  number,  the  value  of  the  fraction  will  not  be  changed. 

EXAMPLES. 

1.  Divide  both  terms  of  |-  by  2  and  by  4. 

2.  Divide  botli  terms  of  \  by  3. 

8.  Divide  both  terras  of  ||  by  2,  3,  4,  6,  and  12. 

4.  Divide  both  terms  of  A|  by  2,  4,  8,  and  16. 

5.  Divide  both  terms  of  ^  by  2,  3,  4,  G,  and  12. 
G.  Divide  both  terms  of  -f-^  by  2,  3,  4,  G,  and  36. 


119.  If  botli  trrins  of  a  fraction  be  divided  by  any  number,  how  will  the 
■  iuo  of  the  fraction  be  ctlectcd  \ 


KEDUOTION    OF   FJi ACTIONS.  117 

REDUCTION  OF  FRACTIONS. 

120.  Reduction  of  Fkactions  is  the  operation  of  cliang- 
in-T  the  fractional  unit  without  alterinjjr  the  value  of  the  fraction. 

A  fraction  is  in  its  loioest  terms,  when  the  numerator  and 
denominator  have  no  common  factor. 

CASE    I. 

121.   To  reduce  a  fraction  to  its  lowest  terms. 

1.  Reduce  y'/'g-  to  its  lowest  terms. 

Analysis. — By  inspection,  it  is  seen  that  5  is      1st.  operation. 
a  common  factor  of  the  numerator  and  denomi-         ^)i-^  —  W' 
nator.    Dividing  by  it,  we  have  ^.    We  then  see 
that  7  is  a  common  factor  of  14  and  35  :  divid-  7)|^  —  ^. 

iug  by  it,  we  have  -|.      Now,  there  is  no  factor 
common  to  2  and  5  :  therefore,  f  is  in  its  lowest  terms. 

2d.  The  greatest  common  divisor  of  70  and  175  is  35.  (Art.  93) ;  if  we 
divide  both  terms  of  the  fraction  by  it,  we  ob-  ^ 

tain,  -S-.     The  value  of  Ihe  fraction  is  not  chang-       2d  operation. 
ed  in  either  operation,  since  the  numerator  and         ^5)^-ts  ~  !• 
denominator  are  both  divided  by  the  same  num- 
ber (Art.  119):  hence,  the  following 

]^ULE. — Divide  the  numerator  and  denominator  hy  their  com' 
mon  factors,  until  they  become  prime  toith  respect  to  each  other. 

Or :  2d.  Divide  the  numerator  and  denominator  hy  their  greatest 
common  divisor. 

exa:\iples. 

Reduce  the  following  fractions  to  their  lowest  terms  : 

6.  Reduce  y|-i. 

7.  Reduce  jVy"^- 


1.  Reduce  -^-^. 

2.  Reduce  j^o- 

3.  Reduce  ^^. 

4.  Reduce  i^4.|. 

5.  Reduce  \y^. 


8    Reduce  ff-hy  2d.  method. 
9.  Reduce  f^f     "         " 


10.  Reduce  V^^   " 


120.  What  is  reduction  of  fractions  1     When  is  a  fraction  in  its  lowest 
terms  1 

121.  How  do  you  reduce  a  fraction  to  its  lowest  terms  1 


118 


REDUCTION    OF    FRACTIONS. 


1 1.  Reduce  y^g^^  by  2(i.  metL 

12.  Reduce  yVaV 

13.  Reduce  ||{}. 
li.  Reduce  j^.%. 
15.  Reduce  f|A|. 


IG.  Reduce  yYA- 


17.  Reduce  f  ^. 

18.  Reduce  jViV 

19.  Reduce  iQff§. 

20.  Reduce  g'^Z/aV 

21.  Reduce  sffo^- 


22.  Reduce  J^V'sV 

CASE    II. 

122,   To  reduce  an  improper  fractio)i  to  an  equivalent  whole  or 

mixed  number. 

1.  In  ^l^  how  many  entire  units  ? 

Analysis. — Since  there  are  5  fifths  in  1  unit,         operation. 
there  will  be  in  278  fifths  as  many  units  1  as  5)278 

5  is  contained  times  in  278,  viz..  55  and  -f  times.  55|-. 

Hence,  the  following 

Rule. — Divide  the  numerator  hy  the  denominator,  and  the  qm 
tient  will  be  the  equivalent  whole  or  mixed  number. 

EXAMPLES. 

Reduce  the  following  fractions  to  whole,  or  mixed  numbers 


1. 

Reduce  \^^. 

9. 

Reduce  ^VrV^  acres 

2. 

Reduce  ^-^. 

10. 

Reduce  \\\\ 

3. 

Reduce  \^*. 

11. 

Reduce  Y-,Y6^- 

4. 

Reduce  ^^T- 

12. 

Reduce  ^|^?|". 

5. 

Reduce  Y/  pounds. 

13. 

Reduce  ''ff ". 

6.' 

Reduce  2||8  ^ays. 

14. 

Reduce  Vgr . 

7. 

Reduce  ^|f  *  yards. 

15. 

Reduce  ='VW- 

8. 

Reduce  "gV/. 

16. 

Reduce  ^'H^''*. 

CASE  in. 
123.    To  reduce  a  mixed  manber  to  an  equivalent   imprope 
fraction. 

1.  Reduce  12|-  to  its  equivalent  improper  fraction. 

122.  What  is  an  improper  fraction  1  How  do  you  reduce  an  irapropev 
fraction  to  its  equivalent  whole,  or  mixed  number  ! 

123  What  iii  a  mixed  number!  How  do  you  reduce  a  mixed  numbei 
to  an  improper  fraction  1  How  do  you  reduce  a  whole  number  to  a  frac- 
tion having  a  given  denominator! 


REDUCTION    OF   FE ACTIONS.  119 

Analysis. — Since  in  any  number 

there  are  7  times   as   many  7tlis  as  operation. 

units  1,  there  will  be  84  sevenths  in  12  x  7  =  84  sevenths. 

12:     To  these  add  5  sevenths,  and  add                    5  sevenths, 

the  equivalent  fraetion  becomes  89  gives    12|=89  sevenths, 

sevenths,     hence,  the  following  Aiis.  =  %p. 

Rule. — Midiipli/  the  ivhole  number  by  the  denominator :  to  the 
2)roducl  add  the  numerator,  and  place  tlie  sum  over  the  given 
denominator. 

EXAIUPLES. 

1.  Reduce  39|-  to  its  equivalent  improper  fraction. 

2.  Reduce  112j?q  to  its  equivalent  improper  fraction. 

3.  Reduce  427ii  to  its  equivalent  improper  fraction. 

4.  Reduce  G7G|y  to  an  improper  fraction. 

5.  Reduce  367^^^  to  an  improper  fraction. 

6.  Reduce  847  y^-^  to  an  improper  fraction. 

7.  Reduce  G7426  |^yf  to  an  improper  fraction. 

8.  How  many  200ths  in  6751^  J  ? 

9.  How  many  151  ths  in  187yVr? 

10.  Reduce  149g-  to  an  improper  fraction. 

11.  Reduce  375^^  to  an  improper  fraction. 

12.  Reduce  17494^|-|f3-  to  an  improper  fraction. 

13.  Reduce  4834|^|-  to  an  improper  fraction. 

14.  Reduce  1789|-  to  an  improper  fraction. 

15.  In  125|-  yards,  how  many  sevenths  of  a  yard  ? 
IG.  In  375|-  feet,  how  many  fourths  of  a  foot  ? 

17.  In  4641^  hogsheads,  how  many  sixty-thirds  of  a  hogs- 
head ? 

18.  In  96Jj'j5-  aci'es,  how  many  640ths  of  an  acre  ? 

19.  In  984y'y'2-  pounds,  how  many  112ths  of  a  pound  ? 

20.  In  353?^  years,  how  many  366ths  of  a  year? 

21.  How  many  one  hundred  and  thirty-fifths  are  there  in  the 
mixed  number  87j'*Jy  ? 

22.  Place  4  sevens  in  such  a  manner  that  they  shall  express 
the  number  78. 

23    By  means  of  5  threes  write  a  number  that  is  equal  to  334. 


120  REDUCTION  OF  FKACTI0N8. 

CASE  IV. 

123.*  To  reduce  a  wliole  number  to  a  fraction  having  a  given 
denominator  : 

1.  Reduce  17  to  a  fraction  of  whicli  tlie  denominator  shall 
be  5. 

Analysis. — There  are  17  times  as  many  operation. 

fifths  in  17  as  there  are  in  1.     In  1,  there  17  X  5  =  85 

are  5  fifths ;  therefore,  in  17  there  are  17  17  =  ^ 
times  5  fifths  or  85  fifths ;  hence, 

Rule. — Multiply  the  whole  number  by  the  denominator,  and 
write  the  product  over  the  required  denominator. 

EXA3IPLES. 

1.  Change  18  to  a  fraction  whose  denominator  shall  be  7. 

2.  Change  25  to  a  fraction  whose  denominator  shall  be  12. 

3.  Change  19  to  a  fraction  whose  denominator  shall  be  8. 

4.  Change  29  to  a  fraction  whose  denominator  shall  be  14. 

5.  Change  65  to  a  fraction  whose  denominator  shall  be  37. 

6.  Reduce  145  to  a  fraction  haviner  9  for  its  denominator. 

7.  Reduce  450  to  a  fraction  having  12  for  its  denominator. 

8.  Reduce  327  to  a  fraction  having  36  for  its  denominator. 

9.  Reduce  97  to  a  fraction  having  128  for  its  denominator. 

10.  Reduce  167  to  a  fraction  whose  denominator  shall  be  89. 

11.  Reduce  325  to  a  fraction  whose  denominator  shall  be  75. 

CASE    V. 

124.    To  reduce  a  compound  fraction  to  a  simple  fraction. 

1.  What  is  the  equivalent  fraction  of  f  of  |-? 

Analysis. — Three-fifths  of  \  is  three  times 
\oi  i^:  1  fifth  of  -^  is  -gij  (Art.  117) :  and  3  opkration. 

times  ^\  is  H  (Art.  1 14) ;  hence,  |  of  ^  -  ^  ^  ^  =  ^ 

hence,  the  following 

123.*  How  Jo  you  reduce  a  vvliole  number  to  a  fraction  having  a  given 
denominator  ! 

124.  \\'hat  is  a  compound  fraction  1  How  do  you  reduce  a  compound 
fraction  to  a  simple  fraction  1 


KEDCrCTION    OF   FKACTIONS.  121 

Rule. — Multiply  the  numerators  together  for  a  new  nurnera^ 
tor,  and  the  denominators  tog e titer  fur  a  new  denominator. 

Note. — 1.    If  there  are  mixed  numbers,  reduce  them   to   their 
equivalent  improper  fractions. 

2.  Cancel  every  factor  common  to  the  numerator  and  denominatpr 
before  multiplying. 

EXAMPLES. 

1.  Reduce  |-  of -|  of  |-  to  a  simple  fraction. 

2.  Reduce  -|  of  |  of  |-  to  a  simple  fraction. 

3.  Reduce  |-  of  f  of  2^  to  a  simple  fraction. 

4.  Change  -|  of  f  of  f  of  3^-  to  a  simple  fraction. 

5.  Change  -f^  of  f  of  |-  of  -f-^  to  a  simple  fraction. 

6.  What  is  the  value  of  i  of  i  of  |  of  12i  ? 

7.  What  is  the  value  of  f  of  |  of  4|  ? 

8.  What  is  the  value  of  ,%  of  7^  of  o-V  ? 

9.  Reduce  -^  of  9|-  of  6f  of  2|  to  a  whole  or  mixed  number- 

10.  Reduce  j\  of -^^  of  21|^  to  a  whole  or  mixed  number. 

11.  Reduce  |-  of  f  of  f  of  y^^^  ^^  T3  ^^  ^  simple  fraction. 

12.  Reduce  yY^  of  j^g-  of  j^^-  of  f  to  a  simple  fraction. 

13.  Reduce  3f  off  of  ^^^-j  of  49  to  a  simple  fraction. 

CASE    VI. 

125.    To  reduce  fractions  of  different  denominators  to  equiva- 
lent fractions  that  shall  have  a  common  denominator. 
1.   Reduce  -|,  |-  and  |-  to  a  common  denominator. 

Analysis. — Multiplying  both  terms 

of  the  first  fraction  by  20,  the  product  operation. 

of  the  olher  denominators,   gives  |5..  2X5x4  =  40   1st  num. 

Multiplying  both  terms  of  the  second  4  x  3  x  4  =  48   2d  num. 

fraction  by  12,  the  product  of  the  other  3  X  .5  X  3  =  45  3d  num. 

denominators,  gives  |§.     Multiplying  3  x  5  X  4  =  60  denom. 
both  terms  of  the  third  by  15.  the  pro- 
duct of  the  other  denominators,  gives  ||-.     In  each  case  both  terms 

125.  How  do  you  reduce  fractions  of  different  denominators  to  equiva- 
lent fractions  having  a  common  denominator  1  Note  1. — What  reductions 
are  first  made  T  2.  When  the  numbers  are  small,  how  may  the  work  be 
done''     3.    How  may  the  work  often  be  shortened"! 


122  REDUCTION    OF    FRACTIONS. 

of  the  fraction  have  been  multiplied  by  the  same  number ;  there- 
fore, the  value  is  not  changed  (Art.  118)  :  hence,  the  following 

]^ULE. — Multiply  the  numerator  of  each  fraction  by  all  the 
denominators  except  its  own,  for  the  new  numerators,  and  all  the 
denominators  together  for  a  common  denominator. 

Nq-j-j. — 1  Before  multiplying,  reduce  to  simple  fractions  when 
necessary. 

2.  When  the  numbers  are  small,  the  work  may  be  performed 
mentally ;  thus, 

1,  1  I  become,  |^,  if,  ^  ;  and  f ,  1,  f  become,  if,  |f,  ff . 

EXAMPLES. 

Eeduce  the  following  fractions  to  common  denominators  : 


1.  Reduce  f ,  51  and  f . 


2.  Reduce  f ,  f ,  \  and  i  of  5. 

3.  Reduce  9i '  4i,  2|  and  f 

4.  Reduce  f ,  l,  f ,  ^  and  2i. 

5.  Reduce  21  of  3,  f ,  f  and  f . 

6.  Reduce  2i  of  31  of  f ,  and 


G3  of  |. 


7.  Reduce  |  of  f  of  -|  and  f 
of  f  of  |. 


8.  Reduce  4§,  21  5iand  6. 

9.  Reduce  51  f,  3i  and  3f . 

10.  Reduce  f"'of  5i  i  of  3i 
and  J  2  of  8^. 

11.  Reduce  61  of  2,  f ,  ^  and  i. 

IS^OTE. — 3.  We  may  often  shorten  tlie  work  by  multiplying  the 
numerator  and  denominator  of  each  fraction  by  such  a  number  as 
will  make  the  denominators  the  same  in  all. 

Reduce  the  following  fractions  to  common  denominators  by 
this  method : 

1.  Reduce  ^,  -j^,  |-  and  f  to  a  common  denominator. 

2.  Reduce  f,  ^y  and  f  to  a  common  denominator. 

3.  Reduce  41,  -f^  and  71  to  a  common  denominator. 

4.  Reduce  10|,  f  and  71  to  a  common  denominator. 

5.  Reduce  Gl.  #  and  71  to  a  common  diMiominator. 

G.  Reduce  ■^-,  ^,  141  and  3|-  1o  a  common  denominator. 

7.  Reduce  -^,  |,  2f  and  If  to  a  common  denominator. 

8.  Reduce  f,  i,  ^%  and  ^  to  a  common  denominator. 

9.  Reduce  W*  f '  ^t  ''^"'^  I  *°  ^  common  denominator. 

''^    Reduce  24,,  51,  -^^^r  'i"^^  "^tV  *"  -"^  common  denominator. 


EEDUCTION    OF   FRACTIONS.  123 

CASE    VII. 

125*.   To  reduce  fractions  io  their  hast  common  denominator. 

The  least  common  denominator  of  two  or  more  fractions  is 
the  number  which  contains  only  the  prime  foctors  of  their 
denominators.  Hence,  before  beginning  the  operation,  reduce 
every  fraction  to  a  simple  fraction  and  to  its  lowest  terms. 

1.  Reduce  |-,  |-  and  |  to  their  least  common  denominator. 

OPERATION. 

(36^4)  X  3  =  27  1st  numerator.  2)4   .  6   .  9 


(36 -=-6)  X  5  =  30  2d  numerator.  3)2   .  3   .  9 

(36  -^  9)  X  4  =  16  3d  numerator.  2.1.3 

2X3X2X3  =  36,  least  common  denominator  : 

therefore,   the  fractions,  reduced    to   their    least    common   de- 
nominator, are 

3  6'        3  6'       '*"^        3  6' 

Hence,  the  following 

Rule. — I.  Find  the  least  common  multiple  of  the  denomina-' 
tors   (Art.  98)  :  this  ivill  be  the  least  common  denominator  of 
the  fractions. 

H.  Divide  the  least  common  denominator  hy  the  denominator 
of  each  fraction,  sejwratehj  ;  multiply  the  quotient  hy  the  nume- 
rator and  place  the  product  over  the  least  common  denominator : 
the  results  will  be  the  new  and  equivalent  fractions. 

EXAMPLES. 

1.  Reduce  I,  ^  and  -^-^  to  their  least  common  denominator. 

2.  Reduce  fj,  |-  and  iy  to  their  least  common  denominator. 
-3.  Reduce  2|,  yV  and  -^-^  to  their  least  common  denominator. 

4.  Reduce  5|,  4:-f^  and  -^^  to  their  least  common  denominator. 

5.  Reduce  8y^,  f  and  3^  to  their  least  common  denominator. 

6.  Reduce  9^,  -^^^  and  r^^  to  their  least  common  denominator. 

125*.  What  is  the  least  common  denominator  of  two  or  more  fractions  1 
How  do  you  find  the  least  common  denominator  of  two  or  more  fractions  J 


124  ADDITION   OF   FRACTIONS. 

7.  Reduce  2-|,  3-j^y  and  y^^  to  their  least  common  denominator. 

8.  Reduce  o^,  |-,  f  and  j-^  to  their  least  common  denominator. 
9.  Reduce  |,  -jy  and  -jg  to  their  least  common  denominator. 

10.  Reduce  Aj-,  7-^^  and  ~  to  their  least  common  denominator. 

11.  Reduce  3^,  6j^  and  1  Jg-  to  their  least  common  denominator. 

12.  Reduce  6|,  Sy^-  and  2-^  to  their  least  common  denominator. 

13.  Reduce  Oyy,  G^^  and  3-3  to  their  least  common  denominator. 

14.  Reduce  j^y,  2^  and  1^''^-  to  their  least  common  denominator. 

15.  Reduce  5|^,  Gj-^,  -^  and  -^  to  their  least  common  de- 
nominator. 

ADDITION  OF  COMMON  FRACTIONS. 

126.  The  Sum  of  two  or  more  fractions  is  a  number  which 
contains  the  unit  1  as  many  times  as  it  is  contained  in  the  frac- 
tions taken  together. 

Addition  op  Fractions  is  the  operation  of  finding  the  sum 
of  two  or  more  fractions.     There  are  two  cases  : 

Is/.  When  the  fractions  have  the  same  unit. 
2d.  When  they  have  different  units. 

CASE   I. 

127.  When  the  fractions  have  the  same  rmit. 

1.  What  is  the  sum  of  i,  f ,  |  and  ^  ? 

Analysis. — In  this  example,  the  unit  operation. 

of  the  fraction  is   1,   and  the  fractional         1  +  3  +  6+3  =  13 
unit  ^.       There    is  1  half  in  the  first.  3        hence,  ^^  =  6-J  sum. 
halves  in  the  second.  6  in  the  third,  and 
3  in  the  fourth  :  hcnce^there  are  13  halves  in  all,  equal  to  6 J. 

2.  "What  is  the  sum  of  1£  and  g£  ? 

Analysis. — The  unit  of  both  fractions  is  operation. 

l£.  In  the  first,  the  fractional  unit  is  ^£,  ^.£  =  f  £ 

and  in  the  second,  ^£.     These  fractional  |-jG  =  ^£ 

units,  being  dilTerent,  cannot  be  expressed.  |£  +  ^£  =  £^  =  £\^. 

126.  What  is  the  sum  of  two  or  more  frartions  ?  What  is  idditijn  of 
fractions  1     How  many  rases  are  there  ?     M'hat  arc  they  1 

127.  How  do  you  add  fractions  which  have  the  same  unit  ? 


COMMON    FRACTIONS. 


125 


in  one  collection.  But  ^£  =  f  jC  and  fjC  =  ^£,  in  each  of  which 
the  fractional  unii  is  -JjG  :  hence,  their  sum  is  |-jC  =  £1-^. 

Note. — Only  ur.its  of  the  same  value,  ickether  fractional  or  integral^ 
can  be  expressed  in  the  same  collection. 

From  the  above  analysis,  we  have  the  following 

Rule. — I.  When  the  /radians  have  the  same  denominator, 
add  their  niunerators^  and  -place  the  sum  over  the  common  de- 
nominator : 

II.  When  they  have  not  the  same  denominator,  reduce  them 
to  a  common  denominator,  and  then  add  as  before. 

JVoTE. — 1.  After  the  addition  is  performed,  reduce  every  result  to 
its  simplest  form ;  that  is,  improper  fractions  to  mixed  numbers,  and 
the  fractional  parts  to  their  lowest  terms. 

2.  It  often  abridges  the  operations  in  fractions  to  reduce  them  to 
f^ieir  least  common  denominators,  before  adding  (Art.  125*.) 


1.  Add  f ,  f,  f  and  f 

2.  Add  f  £,  f  £,  f  £  and  ii-£. 

3.  Add  $ i-L ,  %fj,  $if  and  $/y 
4-    Add  1^-   i^   i^   and  -^- 

5.  Add  f ,  f ,  f,  -If  and  Jf . 

6.  Add  ^2,  ^2,  3?2 '  T2  '^"t^  f  1 

7.  Add  i  f  and  ^^. 

8.  Add  f ,  I,  I  and  ,\. 


EXAMPLES. 

9.  Add  f ,  f ,  f  and  ^\. 
10.  Add  I,  j^,  ^  and  fl. 

1 1      Add    7      7      13     J  1  „„,]  1  9 

19    Arid  3    5      9       5    „,-,/i  ]  5 
J. 4.  j^\.aa  -J,  -g-,  jg,  g^  ana  g^. 

13.  Add  j\,  f,  f  and  |. 

14.  Add  1  41  and  |. 

15.  Add  y\,  j5-^,  -2^  and  |. 

16.  Add  j%  fV  I  and  l.* 


Note. — Reduce  each  fraction  to  its  least  common  denominator  be- 
fore adding. 

17.  What  is  the  sum  of  }  and  }  ? 

Analysis. — Reducing  to  a  common 
denominator,  we  find  the  fractions  to 
be  ^  and  /^,  and  their  sum  to  be 
H-     That  is, 

128.   The  sum  of  two  fractions  whose  numerators  are  each  1,  is 
equal  to  the  sum  of  their  denominators  divided  by  their  product. 


OPERATION. 


128.  What  is  the  sura  of  two  fractions  equal  to  when  each  numerator 
is  equal  to  1  1 


i2G  ADDITION    uF 

18.  What  is  the  sum  of  i  and  i?  of  i  and  i?  of  1  and  i? 
of  1  and  yV  ? 

19.  What  is  the  sum  of  yV  and  ^\  ?  of  yV  and  jL?  of  i  and 
1?  of  i.  audi? 

20.  What  is  the  sum  of  12|,  llf  and  lof  ? 

OPERATION. 

IVi^o/e  Numbers.  Fractions. 

12  +  11  +  15  =  38  14-1^-1-4— -63_ 4- _7_Q_  1  _7  5 208_iJL03  . 

then,  38 +  11^3  =  391  o|.  ^,,5 

Note. — When  there   arc  mixed   numbers,  add  the  whole  numbers 
and  the  fractions  separately,  and  then  add  their  sums. 

Find  the  sums  of  the  following  fractions  : 


27.  Add  f ,  j%  of  j\  of  8  and  21. 

28.  Add  4f,  ,»j- of  ^  of  151. 
29.Add3f,  4f  andl6yV 
SO.AddSf,  4|  and  ^  of  16. 

31.  Add  6|,  13f,  181  and  1321. 

32.  Add  124,  201!-,  and  40 


I  8' 


21.  Add  If,  31  and  1  of  7. 

22.Add3ff,7f,iland2ii. 

23.  Add  2f ,  41  and  f  of  5^\. 

24.  Add  12f ,  9f,  f  of  61 

25.  Add  j\  of  61  and  f  of  71. 

26.  Add  1  of  9f  and  |  of  4f. 

33.  Bought  a  cord  of  wood  for  2|-  dollars ;  a  barrel  of  flour 
for  $9^^ ;  and  some  pork  for  $5^ :  what  was  the  entire  cost  ? 

34.  A  person  travelled  in  one  day  351  miles;  the  next,  28^ 
miles  ;  and  the  next,  25^^^  miles  :  how  many  miles  did  he  ti-avel 
in  the  three  days  ? 

35.  A  grocer  bought  4  firkin.s  of  butter,  weighing  re.'pectively 
54|,  55f,  51  j'^^  and  50|4  pounds  :  what  w.as  their  entire  weight  ? 

36.  I  paid  for  groceries  at  one  time  j^  of  a  dollar;  at  ano- 
ther, 31  dollars ;  at  another,  7f  dollars ;  and  at  another,  5^ 
dollars  :  what  was  the  whole  amount  paid  ? 

37.  A  merchant  had  three  pieces  of  Irish  linen  ;  the  first 
piece  contained  22^-  yards ;  the  second  201  yards  ;  and  tlie 
third  21^-  yards  :  how  many  yards  in  the  three  pieces  ? 

38.  A  man  sold  5  loads  of  hay ;  the  first  weighed  18j^C'r^. ; 
the  second  19^1c(6'A  ;  tlie  third  19-^f;w/.;  the  fourth  21  J-ltvt'/. ;  and 
the  fifth  2{^\^cw(. :  what  was  the  weight  of  the  whole  ? 


CX)MMON  FKACTIONS.  127 

39.  A  farmer  has  three  fields ;  the  first  contains  17f  acres ; 
the  second  25^j  acres;  and  the  third  46 j^^  acres:  how  many- 
acres  in  the  three  fields  ? 

40.  A  man  sold  112f  bushels  of  wheat  for  2504  dollars  ;  O/g 
bushels  of  corn  for  62|  dollars;  225^^  bushels  of  oats  for  104^ 
dollars  :  how  many  bushels  of  grain  did  he  sell,  and  how  much 
did  he  receive  for  the  whole  ? 

CASE   II. 

129.    When  the  fractions  have  different  units. 

1.  What  is  the  sum  of  ^Ib.  and  ^oz.  ? 

Analysis. — In  ^!b.  there  are  operations. 

^z.  (Art.  41).  Then,  the  units  ^Ib.  =  f  x  IGoz.  =  ^^oz. 

of  the  fractions  being  the  same,  ^^  oz.  +  -^oz.  =  -^^oz.  -\-  ^oz. 
viz.,  \oz..  we  reduce  to  a  com-  =  -^-^oz.  =  IZ^oz 

mon  denominator  and  add,  and 
obtain  IS^oz. 

Second    Method. —  Three-  ^oz.=\x^Jh.=-^^lh. 

fourths  of  an  ounce  is  equal  to  f''j-  +  ^V^-=l~2f^^-+F57'^-=lli' 
^Ib.  (Art.  41).    Then,  by  add- 
ing, we  iind  the  sum  to  be  ^^Jb.=  \2^'^oz.=  \Zoz.  %\dr . 

Third  Method.— Find  the  {lb.=^xi&oz.=^oz.  =  Vloz.\2\dr. 
value  of  each   fractional    part    ^oz.  =  \x\Gdr.  —  ^dr.=z  12 

in    terms   of   integers    of    the  Sum   -     -     -     -      13        8^ 

lower  denominations,  and  then  add. 

Rule. — Reduce  the  given  fractions  to  the  same  unit,  and  then 
add  as  in  Case  1. 

Or:  Reduce  the  fractions  sejjaratehj  to  integers  of  lower  de- 
nominations, and  then  add  the  denominate  numbers. 

EXAMPLES. 

1 .  Add  f  of  a  yard  to  -|  of  an  inch. 

2.  Add  together  ^  of  a  week,  \  of  a  day,  and  i  of  an  hour. 

3.  Add  ^cwt.,  ^Ib.,  13os'.,  ^cwt.  and  6/i.  together. 

4.  Add  -g^  of  a  pound  troy  to  i  of  an  ounce. 


129.  How  do  you  add  fractions  when  they  have  different  units  1 


128  ADDITION    OF   FRACTIONS. 

5.  Add  ^  of  a  ton  to  -j^^  of  a  hundred  weight. 

6.  Add  ^  of  a  chaldron  to  ^  of  a  busheL 

.  7.  What  is  the  sum  of  f  of  a  tun,  and  f  of  a  hogshead  of 
wine  ? 

8.  Add  1-  of  f  of  a  common  year,  f  of  |  of  a  day,  and  I  of 
I  of  I-  of  191  hours,  together. 

9.  Add  I"  of  an  acre,  f  of  19  square  feet,  and  |-  of  a  square 
inch,  together. 

1 0.  What  is  the  sum  of  i  of  a  yard,  i  of  a  foot,  and  \  of  an 
inch  ? 

11.  What  is  the  sum  of  f  of  a  £,  and  f  of  a  shilling  ? 

12.  What  is  the  sum  of  i  of  a  week,  i  of  a  day,  i  of  an 
hour,  and  f  of  a  minute  ? 

13.  Add  together  |-  of  a  mile,  f  of  a  yard,  and  '  of  a  foot. 

14.  What  is  the  sum  of  f  of  a  leap  year,  ^  of  a  week,  and 
i  of  a  day  ? 

15.  Add  -i  of  a  ton  to  |-  of  a  hundred  weiofht. 

16.  Add  ^Ib.  troy,  J02.  and  ^pivt. 

17.  Add  together  -^^  of  a  circle,  3|-  signs,  |^  of  a  degree,  and 
^  of  5i  minutes. 

18.  What  is  the  sum  of  ^ijcL,  |  of  ^q?:  and  S^na.  ? 

19.  Add  j^g  of  a  cord,  f  cubic  feet,  and  |-  of  1  of  24|-  cubic 
feet. 

20.  What  is  the  sum  of  f  of  i  of  4  cords,  ^^  of  -^^  of  15  cord 
feet,  and  ^  of  31^  cubic  feet  ? 

21.  Add  I  of  3  ell  English  to  j^^  o^  a  yard.' 

22.  Add  together  A  of  3.-J.  IE.  20P.,  |  of  an  acre,  and  |  of 
3E.  15P 

23.  What  is  the  sum  of  ^^  of  a  ton,  -^  of  a  cwt.,  and  -^  of 
an  ounce  ? 

24.  Wliat  is  the  sum  of  •?,-  of  ^  of  a  mile,  ?  of  a  furlon"-,  -'*- 
of  a  rod  and  -^  of  a  foot  ? 

25.  AVliaL  is  ilie  sum  of -^'^  of  a  common  year,  ^^  of  a  week, 
3  of  a  day,  and  f  of  an  hour  ? 


8UETRACT10N    OF    FKACTI0N3.  129 


SUBTRACTION. 

130.  The  diiference  between  two  fractions  is  such  a  number 
as  added  to  the  less  will  give  the  greater. 

SuBTRA-CTiON  of  Common  Fractions  is  the  operation  of  finding 
the  ditl'erence  between  two  fractional  numbers.  Thei'e  are  two 
cases : 

1.  When  the  fractions  have  the  same  unit:  2d.  When  the 
fractio7is  have  different  units. 

CASE   I. 

131.  When  the  fractions  have  the  same  unit.. 
1.  What  is  the  difference  between  ^  and  \  ? 

4  4 

Analysis. — The  unit  of  both  fractions  is  the         operation. 
same,  being  the  abstract  unit  1.    The  fractional         f  —  ^  =  f . 
luiit  is  also  the  same,  being  ^  in  each ;  hence, 
the  diiference  of  tlie  fractions  is  equal  to  the  difference  of  the  frac- 


2. 

4' 


tional  units,  which  is 

2.  "What  is  the  difference  between  ^Ib.  and  f  of  a  pound  ? 

Analysis. — The  unit  in  both  fractions  is  operation. 

lib.     The  fractional  unit  of  the  first  is  \1h.     |— |=i|_l-^=^y6. 
and  of  the  second    ^Ib.     Reducing   to    the 

Bame  fractional  unit,  we  have  ^fZ6.  and  ^^/6.,  the  difference  of 
which  is  -fjlb.  ;  hence, 

Rule  I. — If  the  fractional  unit  is  the  same  in  both,  subtract 
the  less  numerator  from  the  greater,  and  place  the  difference  over 
the  common  denominator. 

II.  When  the  fractional  units  are  different.,  reduce  to  a  common 
denominator  :  then  subtract  the  less  numerator  from  the  greater, 
and  place  the  difference  over  the  common  denominator. 


130.  What  is  the  difference  between  two  fractions  1     What  is  Subtrac- 
tion of  Common  Fractions  1  How  many  cases  are  there  !     What  are  they  ? 

131.  How  do  3'ou  make  the  subtraction  when  the  fractions  have  the 
same  unit  1 


130  SUBTRACTION    OF 

Note. — Reduce  each  fraction  to  a  simple  form   and   to   its   lowest 
terms  before  reducing  to  a  common  denominator. 


EXAMPLES. 


1.  From  1^  take  1. 

2.  From  f f  take  {J-. 

3.  From  if  take  if. 

4.  From  ^^i  take  i^f 

3  0a  305 

5.  From  |  take  |-. 

6.  P>om  ji  take  J-f 

7.  From  ||  take  ||. 

8.  From  37|i  take  \  of  5^. 


9.  From  |  take  ^. 

10.  From  |  take  j\. 

11.  From  25  take  \}. 

12.  From  ^"^  of  3  take  1  of  |. 

13.  From  i  of  f  of  7  take  ^. 

14.  From  3f  take  f  of  ^. 

15.  From  f  of  15  take  |  of  3. 

16.  From  7^  of  2  take  -}  of  -|. 


17.  To  ^vliat  fraction  must  I  add  f  that  the  sum  may  be  -I  ? 

18.  AVhat  number  added  to  1 1  will  make  5  ? 

19.  What  number  is  that  to  which  if  1^  be  added  the  sum 
will  be  17|-? 

20.  From  the  sum  of  3f  and  10|  take  the  difference  of  25i 
and  17^. 

21.  What  number  is  that  from  which  if  you  subtract  i  of  j 
of  a  unit,  and  to  the  remainder  add  -|  of  -g-  of  a  unit,  the  sum 
will  be  9  ? 

22.  If  I  buy  f  of  1^  of  a  vessel,  and  sell  ^  of  ^  of  my  share, 
how  much  of  the  whole  vessel  have  I  left  ? 

23.  A  man  bought  a  horse  for  i  of  4  of  j^  ^^  $500,  and  sold 
him  again  for  |  of  I-  of  ^  of  $1680 :  what  did  he  gain  by  the 
bargain  ? 

24.  Bought  wheat  at  1|-  dollars  a  bushel,  and  sold  it  ibr  2^ 
dollars  a  bushel :  what  did  I  gain  on  a  bushel  ? 

25.  From  a  barrel  of  cider  containing  31^  gallons,  12f  gal- 
lons were  drawn  :  how  much  Avas  there  left  ? 

26.  Bought  10|  cords  of  wood  at  one  time,  and  24|-  cords  at 
another;  after  using  IGJy  cords,  how  much  remained? 

27.  A  merchant  bought  two  firkins  of  butter,  one  containing 
^^To  pounds,  and  the  other  56}^-  pounds ;  he  sold  43||  poundd 
at  one  time,  and  34^  pounds  at  another  :  how  much  had  he  left? 


COMMON   FKACTIONS.  ]31 

28.  A  man  having  $501  expended  $15^^^  for  dry  goods,  and 
$12|-  for  groceries  :  liow  much  had  he  left  ? 

29.  A  boy  having  ^  of  a  dollar,  gave  -i  of  a  dollar  for  an 
inkstand,  and  }  of  it  for  a  slate  :   how  much  had  he  left  ? 

30.  Bought  two  pieces  of  cloth,  one  containing  27f  yards, 
the  other  321-  yards,  from  which  I  sold  4lO\^  yards  :  how  much 
had  I  left  ? 

132.   1.  ^yhat  is  the  difference  between  i  and  i  ? 

Analysis. — Reducing  both  fractions  to  operation. 

a  coinmou  denominator  and  subtracting,       i  —  1  =  -^  —  -j^^  =  -^, 
we  find  the  difference  to  be  -^  ;  that  is, 

The  difference  behveen  two  fractions,  each  of  ivhose  numerators 
is  1,  is -equal  to  the  difference  of  the  denominators  divided  by 
their  product. 


2.  From  i  take  ^. 

3.  From  jL  take  -jL.. 


4.  From  -jL  take  ^. 

5.  From  Jy  take  Jg. 


133.   1.  What  is  the  difterence  between  IQ>\  and  3i  ? 

Analysis. — Since  we  cannot  take  -^-^  from 
■j^.  wc  borrow  1  -.=  \^  from  the  wliole  nuin-  operation. 

her  of  the  minuend,  wliich  added  to  j^^-,  gives  16^^=  16A- 

W :  then  -j^  from  ^|  leaves  if.     We  must  3|-  =    3y\ 

now  carry  1  to  the  next  figure  of  the  subtra-  12W 

hend,  and  say  4  from  16  leaves  12.     Hence, 
to  subtract  one  mixed    number   from   another, 

Subtract  the  fractional  part  from  the  fractional  part,  and  the 
integral  'part  from  the  integral  part. 

1.  What  is  the  diifei-ence  between  144  and  12y^jy? 
2    What  is  the  difference  between  115|  and  391- ? 

3.  What  is  the  difference  between  78-i^  and  A^  ? 

i  h  3  2 

4.  What  is  the  difference  between  48y^j^  and  41l|-  ? 

5.  What  is  the  difference  between  287^V  and  104  -M^. 


132.  What  is  the  difference  between  two  fractions  whose  numerators 
are  each  1  \ 

133.  How  do  you  subtract  one  mixed  number  from  another? 


132  BUBTKACTION    OF 

CASE   II. 

134.    When  the  fractions  have  different  iniits. 

1.  What  is  tlie  difference  between  i  of  a  £  and  ^  of  a  shil- 
ling ? 

OPERATION. 

Analysis. — Reducing  to  the  com-       ■|£  =  |  x  205.  =  ^s. 
mon  unit  \s.^  we  find  the  difference      "^-^s.  —  ^s.  =  ^^s—  ^s.  =  ^s. 
to  be  ^s.  =  9s.  8d.  =  On.  8d. 

Second  Method. — Reducing  to  the      ^■^-  —  3  ^  it^  ~  ^eV- 
common  unit  IjG,  we  find  the  differ-       2^~  eV-^  ~  %q^  ~  i^^- 
ence  to  be  |f  £  =  95.  8^.  =  |f £  =  95.  8d. 

Third  Method. — Reduce  the  frac-       ^£  =  IO5. 
tions  to  integral  units,  and  then  sub-       ^5.  =  Ad. 

tract  as  in  denominate  numbers.  95.     8c?. 

Rule. — Reduce  the  fractions  to  the  same  unit,  and  then  sub- 
tract as  in  Case  1. 

Or :  Find   the  value  of  each  fraction  in  units  of  loiuer  da- 
nominatiom,  arid  then  suht7-act  as  in  denominate  numbers. 

EXAMPLES. 

1.  From  f  of  a  pound  troy  take  ^  of  an  ounce. 

2.  From  |  of  a  ton  take  |-  of  |  of  a  pound. 

3.  Fi'om  |-  of  y  of  a  hogshead  of  wine   take  ^  of  |  of  a 
quart. 

4.  From  >-  of  a  lea2;ne  take  f  of  a  mile. 

5.  What  is  the  difference  between  Ifs.  and  f  of  l\d.  ? 

G.  What  is  the  difference  between  ^1  of  a  degree  and  ~  of 
\  of  a  degree  ? 

7.  From  f|-  of  a  squai-c  mile  take  3GJ  acres. 

8.  From  ^  of  a  ton  take  |  of  12cwt. 


9.  From  l^lb.  troy  take  \  of  an  ounce. 

10.  From  2g-  cords  take  f  of  a  cord  foot. 

11.  From  -^  of  a  yard  take  |  of  an  inch. 

131.  How  do  vou  subtract  w\ien  the  fractions  have  different  units  1 


COMMON  FE ACTIONS.  133 

12.  From  i  of  f  of  a  pound  take  |  of  ^  of  a  dram,  apothe- 
caries' weiglit. 

13.  A  pound  avoirdupois  is  equal  to  14:0Z.  Il2nvt.  IGf/r.  troy, 
what  is  the  difference,  in  troy  weight,  between  the  ounce  avoir- 
dupois and  the  ounce  troy  ? 

MULTIPLICATION  OF  FRACTIONS. 

135.  Multiplication  of  Fractions  is  the  operation  of  taking 
one  number  as  many  times  as  there  are  units  in  another,  when 
one  or  both  are  fractional. 

1.  If  1  pound  of  tea  cost  |  of  a  dollar,  what  will  f  of  a 
pound  cost  ? 

Analtsis. — The  cost  »,"ill  be  equal  to  operation. 

the  price  of  unity  taken  as  many  times  as     $|  X  y  =  ^^^  :=  $3^' 
there  are  luiits  in  the  quantity  (Art.  75). 

One-seventh  of  a  pound  of  tea  will  cost  one-scvcnth  as  much  as  1/6. 
Since  lib.  cost  S|.  \  of  1/6.  will  cost  |  of  Sf^SgV  (Art.  117). 
But  3  sevenths  of  1/6.  will  cost  3  times  as  much  as  \\  that  is, 
Sj^g  X  3  =  $|f  (Art.  114).  Hence,  to  multiply  one  fraction  by 
another, 

Rule. — Multiply  the  numerators  together  for  a  new  numera- 
tor^ and  the  denominators  tor/ether  for  a  new  denominator. 

Notes. — 1.  When  the  multiplier  is  less  than  1,  we  do  not  take  the 
whole  of  the  multiplicand,  but  only  such  a  part  of  it  as  the  multi- 
plier is  of  1. 

2.  When  the  multiplier  is  a  proper  fraction,  multiplication  does 
not  imply  increase.,  as  in  the  multiplication  of  whole  numbers.  The 
product  is  the  same  part  of  the  multiplicand  which  the  multiplier 
is  of  1. 


135.  What  is  multiplication  of  fractions  ]  How  do  you  multiply  ono 
fraction  by  another  ?  When  the  multiplier  is  less  than  1,  what  part  of  the 
multiplicand  is  taken  1  If  the  fraction  is  proper,  does  multiplication  imply 
increase  ?  What  part  is  the  product  of  the  multiplicand  1  What  do  you 
do  when  either  factor  is  a  whole  number  1 


134: 


MULTIPLICATION    OF 


3.  If  either  of  the  factors  is  a  whole  number.  ^ATite  1  under  it  for 
a  denominator. 

4.  ^Yhen  either  of  the  factors  is  a  mixed  number,  it  maybe  reduced 
to  an  improper  traction,  or  we  may  multiply  the  parts  separately  and 
take  their  sum. 


1.  Multiply  I  by  8. 

2.  Multiply  fg  by  12. 

3.  Multiply  ll  by  9. 

4.  Multiply  1^  by  15. 

5.  Multiply  ^^1  by  12. 


6.  Multiply  I  of  i-  by  35. 

7.  Multiply  3i  off  by  14. 

8.  Multiply  If  of  2^  by  16. 

9.  ]\ruhii)ly  2i  of  f  by  70. 
10.  Multiply  4|  of  8  by  36. 


11.  Multiply  36  by  4i. 

Analysis. — The  number  3(5  is  to  be  taken 
4 J  times;  that  is,  4  times  and  i  times.  One- 
ninth  of  36  is  4,  which  is  wTitten  in  the  units 
place :  then.  4  times  36  is  144;  and  the  sum 
148  is  the  product. 


OPERATION. 

36 

4 
144 


148  Ans. 


12.  Multiply  67  by  9^^. 

13.  Multiply  842  by  71-. 

14.  Multiply  360  by  12f. 


15.  Multiply  460  by  llf. 

16.  Multiply  620  by  lOf. 

17.  Multiply  1340  by  8f. 


1.  Multiply  ^  by  8. 

2.  Multiply  15  by  f 

3.  Multiply  11  hyj%. 

4.  Multiply  7|  by  8. 

5.  Multiply  9^  by  18f. 

6.  INIultiply  32-  by  411. 

7.  Multiply  Vr-  by  9. 

8.  Multiply  f  by  I-. 

9.  Multiply  1  by  f . 

10.  Multiplyloff  by  ^. 

11.  Multiply -iV  by /o  of  A- 


EXAJrrLES. 

12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 


Multiply  i  of  1  by  A  of  T^. 

INIultiply  I-  by  16. 

iNIultiply  28  by  -j^. 

:\rultiply  ll-  by  18. 

IMultiply  8^0  by  15. 

Multiply 

IMultiply 

Multi])ly  8421  by  7^. 

ISrultiply 

Multiply 

Multiply 


r. 
TT 


of  2-  by  ^J. 


51  by  A  of  31. 


f\  bv  ^-. 

9    '•V     7 


TTJ 


by  7^ 


TT* 


^?  by  -J  of  ^g. 


COMMON    FRACTIONS.  '  135 

23.  Multii)ly  -j\,  |f  and  ||  together. 

24.  Multiply  if,  ^^,  -^j  and  f{i  together. 

25.  What  is  the  product  of  |^  by  |  of  17  ? 
2G.  What  is  the  product  of  6  by  f  of  5  ? 

27.  AVhat  is  the  product  of  i  of  i  of  3  by  loj  ? 

28.  Require  the  product  of  f  of  |-  by  f  of  3|. 

29.  Require  the  product  of  5,  ^,  |  of  |,  and  41-. 

30.  Require  the  product  of  14,  f ,  ^  of  9,  and  6f . 

31.  What  A\ill  7  yards  of  cloth  cost  at  $f  a  yard  ? 

32.  Wliat  will  12|-  bushels  of  apples  cost  at  $|  a  bushel  ? 

33.  If  one  bushel  of  wheat  cost  $1^,  what  will  f  of  a  bushel 
cost  ? 

34.  If  one  horse  eat  f  of  a  ton  of  hay  in  one  month,  how 
much  will  18  horses  eat  in  the  same  time  ? 

35.  If  a  man  earn  $i|  in  one  day,  how  much  can  he  earn  in 
24  days  ? 

36.  What  will  31  yards  of  cloth  cost  at  |-  of  a  dollar  a  yard  ? 

37.  At  $16  a  ton,  what  will  |i  of  a  ton  of  hay  cost  ? 

38.  If  one  pound  of  tea  cost  $1\,  what  will  61  pounds  cost? 

39.  What  will  3|  boxes  of  raisins  cost  at  $2^  a  box  ? 

40.  At  75  cents  a  bushel,  what  will  ii  of  a  bushel  of  corn 
cost  ? 

41.  If  a  lot  of  land  be  worth  $75y^3-,  what  will  ^x  of  it  be 
worth  ? 

42.  If  a  man  earn  $56  in  one  month,  how  much  can  he  earn 
in  y-j  of  a  month  ? 

43.  What  will  171  yards  of  cambric  cost  at  21  shillings  a 
yard  ? 

44.  Bought  15|-  barrels  of  sugar  at  $20|  a  barrel,  what  did 
the  whole  cost  ? 

45.  If  one  bushel  of  corn  is  worth  f  of  a  dollar,  what  is  f  of 
a  bushel  worth  ? 

46.  If  I  own  -^j  of  a  farm  and  sell  -^^  of  my  share,  what 
pai-t  of  the  whole  farm  do  I  sell  ? 

47.  Bought  a  book  for  -j^  of  a  dollar  and  a  knife  for  j^  as 
much;  how  mucli  did  I  pay  for  the  knife  ? 


136  MDLTIPLICATION 

48.  At  f  of  |4  of  a  dollar  a  pound,  what  will  f  of  ||-  of  a 
pound  of  tea  cost  ? 

40.  If  ha  J  is  Avorth  $9|-  a  ton   wliat  is  ^  of  Si  ton  worth. 

50.  If  a  man  can  dig  a  cellar  in  22^  days,  how  many  days 
will  it  take  him  to  dig  ^  of  it  ? 

51.  If  a  railroad  train  run  1  mile  in  -^L.  of  an  hour,  how 
long  will  it  be  in  running  106j  miles  ? 

52.  What  will  be  the  cost  of  20|-  cords  of  wood  at  $3J  a 
cord  ? 

53.  If  a  man  walk  3^  miles  an  hour,  how  far  will  he  walk  in 
Of  hours  ? 

54.  "What  will  14|  bushels  of  potatoes  cost  at  311  cents  a 
bushel  ? 

55.  What  will  12J  dozens  of  eggs  bring  at  18|-  cents  a 
dozen  ? 

56.  At  "I  of  a  dollar  a  bushel,  what  will  1021  bushels  of  ry8 
cost? 

57.  What  will  |-  of  a  firkin  of  butter  cost  at  $18Jg-  a 
firkin  ? 

58.  A  man,  at  his  death,  left  his  wife  $15000;  she  at  her 
death  left  |-  of  her  share  to  her  daughter :  what  part  of  the 
father's  estate  did  the  daughter  receive  ? 

59.  A  person  owning  ^  of  a  cotton  factory  sold  f  of  his  paii; 
to  A,  and  the  rest  to  B :  what  jiart  of  the  whole  did  each 
buy? 

GO.  A  owTied  I-  of  a  farm  and  sold  4  of  his  share  to  B,  who 
sold  f  of  what  he  bought  to  C,  who  sold  £  of  what  he  bought 
to  D :  what  part  of  the  whole  did  D  have  ? 

61.  A  owned  |  of  200  acres  of  land,  and  sold  ^  of  his  sliare 
to  B,  who  sold  i  of  what  he  bought  to  C  :  how  many  acres  had 
each? 


DIVISION   OF   FRACTIONS.  137 

DIVISION. 

136.  Division  of  Fractioxs  is  tlie  operation  of  finding 
a  number  wliicli  multiplied  by  the  divisor  will  produce  the 
dividend,  when  one  or  both  are  fractional. 

1.  What  is  the  quotient  of  |-  divided  by  -J^*-  ? 

Analysis. — How  many  lanes  is  ^^  con-  operation. 

tallied  in  |-?     If  ^  be  divided   by  14,  the  l-^XA^l-^J^ 

quotient  will  be  gx. .    (Art.  117).     Since  ~  iVa' ~  1^6    -^^'* 

the  true  divLsor  is  but  ^  of  14,  the  divisor 
used  is  5  times  too  large  ;  hence,  the  par- 
tial quotient  r^^^  is  5  times  too  small. 
Multiplying  this  by  5,  we  have  the  true 
quotient,  ,3_5^  =  ■^\.  2     ^      ! 

The  operation  may  be  made   by  means  X4     0 

of  the  vertical  line,  by  .simply  placing  the 


dividends  on  the  right  and  the  divisors  on  16  |  0  —  -^^  A. 

the  left. 

Since  the  same  process  is  applicable  to  any  two  fractions, 
we  have  the  following  rule  : 

I.  Invert  the  terms  of  the  divisor : 

II.  Multiply  the  numerators  together  for  the  numerator  of  the 
quotient,  and  the  denominators  together  for  the  denominator  of 
the  quotient. 

Notes. — 1.  If  either  the  dividend  or  divisor  is  a  whole  number. 


make  it  fractional,  by  writing  1  under  it  for  a  denominator. 

2.  If  the  vertical  line  is  used,  the  denominator  of  the  dividend  and 
the  numerator  of  the  divisor  fall  on  the  left,  and  the  other  terms  on 
the  right.  . 

3.  Cancel  all  common  factors. 

136.  What  is  division  of  fractions  ^  AMiat  is  the  rule  for  the  division 
effractions'!  What  do  you  do  when  cither  the  dividend  or  divisor  is  a 
whole  number  1  Where  do  the  parts  fall  when  you  use  the  vertical  line  '^ 
What  do  you  do  when  either  term  of  the  fraction  is  a  mixed  number  or 
a  compound  fraction  1  If  the  terms  of  the  dividend  are  exactly  divisible 
l)y  the  corresponding  terms  of  the  divisor,  how  do  you  find  the  quotient  ? 

7 


L38 


DIVISION   OF 


4.  If  the  dividend  and  divisor  have  a  common  denominator,  it 
will  cancel,  and  the  quotient  of  the  numerators  will  he  the 
answer. 

5.  When  either  term  of  the  fraction  is  a  mixed  number,  or  a  com- 
pound fraction,  reduce  to  the  form  of  a  simple  fraction  before 
dividing. 

6.  If  the  numerator  of  the  dividend  is  exactly  divisible  by  the 
numerator  of  the  divisor,  and  the  denominator  by  the  denominator, 
the  division  may  be  made  without  inverting  the  terras  of  the  divisor. 

EXAMPLES. 


1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
lo. 
16. 
17. 
18. 
10. 
20. 
21. 
22. 
23. 
24. 
25. 


Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 


H  by  7. 
t\  by  6. 

If  by  9. 
m  by  40. 
M  by  13. 

5by  tV- 
27  by  I 

^byf 


fVby 
^by 
off  by  f  off. 


5 

TT* 


I  off  by. 


*  of  ^ 


I  Of  f  by  f  off. 
56  by  11. 
1000  by  T^. 
725  by  f  f 
4f  by  5. 
9^5^  by  12. 
of  16iby  41. 
by  1  of  7.  ' 
^  of  50"  by  41. 
300^'V  by  61. 
1  of"3ii  by  l>f  71 
91  by  81. 


1 

h 


»     OI    yy 


by  6^. 


26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. 
34. 
35. 
36. 
37. 

oo 
OO. 

39. 
40. 
41. 
42. 
43. 
44. 
45. 
46. 
47. 
48. 
49. 
50. 


Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 
Divide 


by  4. 


_9 

ro- 


il 

il  by  5. 
ffby8. 
m  by  48. 
tS's  by  21. 
36  by 
420  by  f 

3 

¥• 

7 
15" 

byM- 
I-  by  if. 

foffbyfoff. 
lofioffbylofl 
650  by  J-^f . 
1273  by  li. 
4324  by 


2'o  by 

25 

2  of  2  7. 

3  "^    5  0 


by 


]  28 
475' 


6f  by  8. 


121 


by  42 
r7  by  91. 
100  by  43 

44^  by 
lll-Vby 

by 

by  I  of  U. 

by  I  of 


191J- 
5205} 


'8' 
2  13 
3'3  3* 

33i. 
1591. 


90 


COMMON    FRACTIONS.  139 

51.  At  J-  of  a  dollar  a  pound,  how  much  butter  can  be  bought 
for  I  k  of  a  dollar  ? 

52.  At  f  of  a  dollar  a  yard,  how  muck  cloth  can  be  bought 
for  |-  of  a  dollar  ? 

5-),  If  a  bushel  of  potatoes  cost  ^  of  a  doljar,  how  many  can 
be  bought  for  -^^  of  a  dollar  ? 

54.  If  ^  of  a  ton  of  hay  will  feed   1   horse  one  week,  how 
many  hoi^^es  will  -f^j  of  a  ton  feed,  the  same  time  ? 

55.  If  ^^  of  a  bushel  of  apples  cost  |  of  a  dollar,  what  will  a 
bushel  cost  ? 

5G.  What  will  a  barrel  of  flour  cost,  if  j^^  of  a  barrel  cost 
I  of  a  dollar  ? 

4 

57.  If  |-  of  a  bushel  of  apples  cost  -|  of  a  dollar,  what  will 
1  bushel  cost  ? 

58.  How  much  molasses  at  f  of  a  dollar  a  gallon,  can  be 
bought  for  ly  dollars  ? 

59.  A  man  sold  |-|-  of  a  mill,  which  was  ^  of  his  share  : 
what  part  of  the  mill  did  he  own  ? 

60.  Yf  hat  number  multiplied  by  f,  will  give  lof  for  the  pro- 
duct ? 

61.  What  number  multiplied  by  51,  will  give  146  for  the  pro- 
duct? 

62.  The  dividend  is  520^,  and  the  quotient  36y%  :  what  is 
the  divisor? 

63.  What  number  is  that  which  if  multiplied  bv  I-  of  ^  of 
151,  will  produce  -I? 

64.  If  lib.  of  sugar  cost  ^j  of  a  dollar,  what  will  1  pound 
cost? 

65.  If  lQ\Ih.  of  nails  cost  y  of  a  dollar,  what  is  the  price  per 
pound  ? 

66.  If  y  of  a  yard  of  cloth  cost   $3,  what  is   the  cost  of  a 
yard  ? 

67.  A  fiuiiily  consumes  165|  pounds  of  butter  in  8^  weeks: 
how  much  do  they  consume  in  1  week  ? 

68.  At  $9-1  a  barrel,   how   much   flour   can  be  bought  for 


$138f ? 


140  REDUCTION    OF 

69.  If  a  man    divides   $3|  equally   among  8  beggars,  how 
much  will  he  give  them  apiece  ? 

70.  If  8  pounds  of  tea  cost  7f  dollars,  what  is  the  price  per 
pound  ? 

71.  If  I  of  a  ton  of  hay  sell  for  $10f,  what  should  1  ton  sell 
for  ? 

72.  If  ^  of  an  acre  of  ground  produce  84y^  bushels  of  pota- 
toes, how  many  bushels  will  1  acre  produce? 

73.  What  quantity  of  cloth  may  be  purchased  for  $5Jg-,  at 
the  rate  of  $6f  a  yard  ? 

74.  How  long  would  a  person  be  in  travelling  125^  miles,  if 
he  travelled  oljj  miles  per  day  ? 

75.  How  many  bottles,  each  holding  1^  gallons,  can  be  filled 
from  a  barrel  of  wine,  containing  311  gallons  ? 

7G.  How  long  will  it  take  11  men  to  do  a  piece  of  work  that 
I  man  can  do  in  15|  days  ? 

77.  If  |-  of  a  barrel  of  flour  cost  G  dollars,  what  is  the  price 
per  barrel  ? 

78.  Eighty-one  is  f  of  how  many  times  8  ? 

79.  Five-eighths  of  48  is  f  of  how  many  times  9  ? 

80.  How  many  times  can  a  vessel,  containing  3-  of  a  gallon, 
be  filled  from  i  of  a  barrel  of  311  gallons  ? 

81.  If  5}Jb.  of  tea  cost  $41-,  what  is  the  price  of  1  pound  ? 

82.  If  f  of  -^   of  a  ship  is  worth  $2540,  what  is  the  whole 
vessel  worth? 

83.  If  |-  of  an  acre  of  land  cost  S361,  Avhat  will  be  the  value 
of  an  acre  ? 

84.  If  I  of  -J  of  a  bai'rel  of  flour  will  last  a  family  1  week, 
how  lonn;  will  9fV  barrels  last  them? 


COMPLKX    FRACTIONS.  141 

REDUCTION  OF  COMPLEX  FRACTIONS. 
137.  A  Complex  Fraction  is  only  another  form  of  expres- 
Eion  for  the  division  of  fractious:  thus, -3,  is    the   same  as   | 

"6 

divided  by  f ;  and  may  be  written,  i^^=is 


5 

o3 

X 

"$ 

7 

138.    To  reduce  a  complex  fraction  to  a  simple  fraction  : 

1.  Reduce  —y-  to  a  simple  fraction. 

Analysis. — Reducing  the  divisor  operation. 

and  dividend  each  to  a  simple  frac-         G|-  =  ^j)     and     1|  =  f-. 

tion,  we  have  %»  and  f .     Then  2_o      20  _:.  8  ^  2j)  x  1  =  83  -  54 
divided  byf  is  equal  to  5/ xX^Jj^a      ^    "^        ''    '    ^        ^' 

_3_5  _   t5 

—  6    —  "^a- 

Rule. — Divide  the  numerator  g  |  35 

of  the  compilcx  fraction  hy  its  de-  Ans.  ^^-  =  5|. 

nomi/iator. 

Or:  Midtiply  the  numerator  of  the  upper  fraction  into  the  de- 
nominator of  the  lower,  for  a  new  numerator  ;  arid  the  denominator 
of  the  upper  fraction  into  the  numerator  of  the  lower,  for  a  new 
denominator. 

]v;oTES. — 1.  When  either  of  the  terms  of  a  complex  fraction  is  a 
mixed  number,  or  a  compound  fraction,  it  must  first  be  reduced  to  the 
form  of  a  simple  fraction. 

2.  ^Yhen  the  vertical  line  is  used,  the  numerator  of  the  upper  and 
Ihe  denominator  of  the  lower  numbers  fall  on  the  right  of  the  vertical 

line,  and  the  other  terms  on  the  left. 


137.  What  is  a  complex  fraction  1 

138.  How  do  you  reduce  a  complex  fraction  to  a  simi)le  fraction  ^ 


142 


APPLICATIONS    m 


EXAMPLES. 


Reduce  the  following  to  simple  fractions 


1.  Reduce  f- 

4 
5 

2.  Reduce  ^' 

T6 

3.  Reduce  -^• 

87i 

4.  Reduce  -y^- 

8 

5.  Reduce  -^• 

84 

6.  Reduce  r^- 


7.  Reduce  ~-^^' 


8.  Reduce 


20 

7 


^  of  7  3- 
9.  Reduce  ^~J~^^- 

26.8 
10.  Reduce 


'3  5 


11.  Reduce 


I  of  17 
554 


H 


12.  Reduce  f  off^  of  -| 


APPLICATIONS    IX    FRACTIONS. 

1.  What  -will  51  cords  of  wood  cost  at  L  of  |  of  |  of  $50  a 
cord  ? 

2.  A  farmer  sold  f  of  a  ton  of  hay  for  $6| :  what  would  be 
the  price  of  a  ton  at  the  same  rate  ? 

3.  A  person  walks  77 1  miles  in  10^  hours  :  at  what  rate 
is  that  per  hour  ? 

4.  From  the  product  of  ^  and  llJ,take  y^,  and  multiply  the 
remainder  by  20'. 

5.  How  mucli  greater  is  ^  of  the  sum  of  ^,  ^,  i  and  J ,  than 
the  sum  of  -^,  i  and  ^  ? 

6.  If  •}  of  a  ton  of  hay  is  worth  $7i,  what  is  2|  tons  worth? 

7.  If  -   of  a  dollar  will  pay  for  |-  of  a  yard  of  cloth,  how 
piany  yards  can  be  bouglit  for  $ll-i?-  ? 

8.  AVhat  is  the  value  of  3J-  cords  of  wood  at  S-l|  a  cord  ? 

4      3-  A 

9.  "What  is  the  continued  product  of  144,    -— ,    v,  ^^^  ^? 

<■  ''    oi'    f  9 

49''-  34?- 

10.  "Wiiat  is  the  sum  and  (lidi'rencc  of  --3  and  vr,rV  ? 

9/  14"TT 


COMPLEX    FRACTIONS.  143 

11.  At  }  of  a  dollar  a  peck,  how  many  busliels  of  apples  can 
be  bought  for  $Gf  ? 

12.  What  is  the  difference  between  f  of  a  league  and  -^  of 
a  mile  ? 

13.  Subtract  8|Z6.  from  |-  of  a  cwf. 

14.  What  is  the  sura  of  A^^  miles,  ^  of  a  furlong,  and  f  of 
11  yards  ? 

15.  At  $lf  per  day,  how  many  days  labor  can  be  obtained 
for$36f? 

16.  Sold  7i  bushels  of  apples  for  $3| :  what  should  I  receive 
for  24f  bushels  ? 

17.  A  has  G34  sheep,  which  are  124  more  than  f  of  21  times 
B's  number  :  how  many  sheep  had  B  ? 

18.  At  3^  of  a  dollar  a  yard,  how  many  yards  of  ribbon 
can  be  bought  for  ^  of  a  dollar  ? 

19.  Paid  $56 1  for  94  yards  of  muslin  :  how  much  was  that 
per  yard  ? 

20.  Bought  51  yards  of  cloth  at  $41  a  yard,  and  paid  for  it 
in  wheat  at  $11  a  bushel :  how  many  bushels  were  required  ? 

21.  What  number  must  be  taken  from  27|,  and  the  remainder 
multiplied  by  14f ,  that  the  product  shall  be  100  ? 

22.  Three  persons.  A,  B,  and  C,  purchase  a  piece  of  pro- 
perty for  $6300;  A  pays  f  of  it,  B,  land  C  the  remainder: 
what  is  the  value  of  each  one's  share  ? 

23.  What  number  is  that  which  being  diminished  by  the  dif- 
ference between  f  and  |  of  itself  leaves  a  remainder  equal  to  34  ? 

24.  Add  together  1  of  a  week,  1  of  a  day,  and  1  of  an  hour. 

25.  What  is  the  sum  of  f  of  £15,  £3f,  i  of  f  of  |  of  £1, 
and  f  of  f  of  a  shilling  ? 

26.  If  1  of  John's  marbles  are  equal  to  i  of  James',  and  to- 
gether they  have  56  :  how  many  has  each  ? 

27.  A  person  owning  f  of  2000  acres  of  land,  sold  f  of  his 
share :  how  many  acres  did  he  retain  ? 

28.  A  boy  having  240  marbles,  divided  them  in  the  following 
manner  :  he  gave  to^A,  l,  to  B,  j\,  to  C,  1,  and  to  D,  i,  keeping 
the  remainder  himself :  what  number  of  marbles  had  each  ? 


144  DUODECIMALS.  , 

29.  A  man  engaging  in  trade  with  $3740,  found  at  the  end 
of  3  years  that  he  had  gained  $156^  more  than  ^  of  his  capital : 
what  was  his  average  annual  gain  ? 

30.  Two  boys  having  bought  a  sled,  one  paying  |-  of  a  dollar, 
and  the  other  |-  of  a  dollar,  sold  it  for  y^g-  of  a  dollar  more  than 
they  gave  for  it :  what  did  they  sell  it  for,  tind  what  was  each 
one's  share  of  the  gain  ? 

31.  A  farmer  having  126y  bushels  of  wheat,  sold  -I  of  it  for 
$21  a  bushel,  and  the  remainder  for  $1|-  a  bushel :  how  much 
did  he  receive  for  his  wheat  ? 

32.  A  man  having  $19-1-,  expended  it  for  wheat  and  corn,  of 
each  an  equal  quantity ;  for  the  wheat  he  paid  $14  a  bushel, 
and  for  the  corn  $|-  a  bushel :  how  much  of  each  did  he  buy  ? 

33.  Two  persons  engage  in  trade :  A  furnished  -^^^  of  the 
capital,  and  B,  -^ ;  if  B  had  furnished  $49 2|-  more,  their  shares 
would  have  been  equal :  how  much  did  each  furnish  ? 

34.  A  man  being  asked  how  many  sheep  he  had,  said  he 
had  them  in  3  fields  :  in  the  first  he  had  G3,  which  was  ^  of. 
what  he  had  in  the  second,  and  that  f  of  what  he  had  in  the 
second  was  just  4  times  what  he  had  in  the  third  :  how  many 
sheep  had  he  in  all  ? 

DUODECIMALS. 

139.  Duodecimals  are  a  system  of  numbers  which  arise  from 
dividing  the  unit  1  according  to  the  uniform  scale  of  12  ;  tlms, 

If  the  unit  1  foot  be  divided  into  12  equal  parts,  each  part 
is  called  an  inch  or  prime,  and  marked  \  If  an  inch  be  divi- 
ded into  12  equal  parts,  each  part  is  called  a  second,  and 
marked  ".  If  a  second  be  divided,  in  like  manner,  into  12 
equal  parts,  each  part  is  called  a  third,  and  marked  '"  ;  and  so 
on  for  divisions  still  smaller. 

139.  What  arc  duodecimals  1  If  the  unit  1  foot  be  divided  into  12  equal 
parts,  what  is  each  part  called?  If  1  inch  be  divided  into  12  equal  parts, 
what  is  each  part  called  ]  If  the  second  be  divided  in  like  manner,  what 
is  each  part  called  1     What  are  indices  1 


DUODECIMALS.  145 

This  division  of  the  foot  gives 

1'     inch  or  prime —  ^-      of  a  foot. 

1"    second  is  Jj  of  jL  -     -     -     -     =  J-^     of  a  foot. 
V"  third  is  tV  of  yV  of  t'2     -     -     =  W^  of  a  foot. 

Note.— The  marks  ',  ",  '",  &c.,  which  denote  the  fractional  units, 
are  called  indices. 


TABLE. 

12'" 

make 

1" 

second. 

12" 

a 

r 

inch  or  prime. 

12' 

(( 

1 

foot. 

Hence :  Duodecimals  are  denominate  fractions,  in  which  the 
primary  unit  is  1  foot,  and  the  scale  uniform,  the  units  of  it,  at 
every  point,  being  12. 

Note. — Duodecimals  are  chiefly  used  in  measuring  surfaces  and 
solids. 

ADDITION  AND  SUBTRACTION. 

140.  The  units  of  duodecimals  are  reduced,  added,  subtracted, 
and  muhiplied  like  those  of  other  denominate  numbers.  The 
units  of  the  scale  are  12,  at  every  change  of  the  unit. 


EXAMPLES. 


1.  In  86'  how  many  feet? 

2.  In  750"  how  many  feet? 

3.  In  37000'"  how  many  ft.  ? 


4.  In  G7'  how  many  feet? 

5.  In  470'"  how  many  feet? 

6.  In  375"  how  manv  feet  ? 


7.  What  is  the  sum  of  8/?.  9'  7"  and  G/>.  7'  3"  4'"? 

8.  What  is  the  difference  between  32/?.  6'  G"  and  20ft.  7'"? 

9.  Add  together  9/^.  G'  4"  3"',  12//.  2'  9"  10"',  26ft.  0'  5", 
and  40/2.  1'  0"  3'". 

10.  Wliat  is  the  sum  of  126ft.  0'  G",  45/?.  11'  0"  2'"  and 
12ft.  G'  ? 


140.   By  what  rules  do  you  operate  on  duodecimal  units  I     What  are  tha 
unils  of  the  scale  ! 


1 4:0  DUODECIMALS. 

11.  What  is  the  sum  of  84//.  7',  96//.  0'  11",  42//.  6'  9'' 
10"'  and  5'  7"  11'"? 

12.  From  127//.  3'  6"  4'"  11""  take  40//.  0'  10"  7"'  5"". 

13.  What  is  the  difference  between  425/7.  9'  10"  and  107//. 
10'  9"  8'"? 

14.  What  is  the  sum  and  difference  of  325//.  7'  6"  2'"  and 
217/.  10'  9"? 

15.  What  is  the  sum  and  difference  of  1001//.  0'  0"  10'" 
and  720//.  10'  9"  1'"  ? 

MULTIPLICATION. 

141.  Multiplication  of  duodecimals  is  the  operation  of 
finding  the  superficial  contents,  or  the  contents  of  volume,  when 
the  linear  dimensions  are  known. 

To  do  this  we  begin  with  the  highest  unit  of  the  multiplier 
and  the  loiuest  of  the  multiplicand,  and  recollect, 

1st.  That  1  linear  foot  x  1  linear  foot  =  1  square  foot,  (Art. 
411),  or,  that  a  part  of  a  foot  x  a  part  of  a  foot  =  some  part 
of  a  square  foot. 

2d.  That  a  square  foot  X  by  a  foot  in  length  =:  a  cubic  foot. 

Note. — Ob.serve  that  in  the  first  muUiplication  the  unit  is  changed, 
from  a  linear  to  n  superficial  unit;  in  the  second  muUiplication,  from 
a  supevficial  unit  to  a  unit  of  volume. 

1.  Multiply  6//.  7'  8"  by  2//.  9'. 

Analysis. — Since  a  prime  is  ^  of  a 
foot,  and  a  second  y^,  2X8"=  ^^  of 
a  square  foot ;  which  reduced  to  12ths, 
is  r  and  A";  that  is,  1  twelfth,  and  4 
twelfths  of  twelfths  of  a  square  foot. 
2  X  7'  =  14  twelfths  =  1ft.  2'    -     -     - 

2  X  6  =  12  square  feet 2X6=12 

9'  X  8"=  ylfg-  of  a  square  foot  ----  6"    -       9'x  8"=  6" 

9'x  T  =  ^^  =  5'  3" 9'X  7'  =         5'  3" 

9  X  G'  =14    =  4  6' 9'X  6   =4  6' 

Prod.     18  3'   1" 


OPERATION, 

ft- 

6 

7' 

8" 

X 

8"  = 

o 

9' 

2 

1' 

4" 

2 

X 

7'  = 

1 

2' 

HI.  Wliat  is  multiplication  of  duodecimals  1 


DUODECIMALS.  147 

Rule. — I.  Write  the  midtijylier  under  the  mtdtij^licand,  so 
that  units  of  the  same  order  shall  fall  in  the  same  column. 

II.  Begin  with  the  highest  unit  of  the  multiplier  and  the 
lowest  of  the  multiplicand,  and  make  the  index  of  each  product 
equal  to  the  sum  of  the  indices  of  the  factors. 

III.  Reduce  each  product  of  the  first  midtijylication  to  square 
feet  and  12ths  of  a  square  foot,  and  when  there  ore  three  factors 
reduce  the  second  products  to  units  of  volume, 

NoTK. — The  index  of  the  unit  of  any  product  is  equal  to  the  sum 
of  the  indices  of  the  factors. 

EXAMPLES. 

1.  How  many  cubic  feet  in  a  stick  of  timber  12  feet  6  inches 
long,  1  foot  5  inches  broad,  and  2  feet  4  inches  thick  ? 

Analysis. — Beginning  with  the  1  foot,  operation. 

we  say  1  time    4'  is   4'  =  ^^  of  a  square  ft. 

foot :  then,  1  time  2    is    2  square  feet.  2     4' 

Next,  5  times  4'  are   20"  =  1'  and   8"  :  1 5^ 

then,  5  times  2  feet  =  10',  and  the  V  to 
carry,  makes  11'  8".  Then  multiplying 
by  the  length  12  feet  6',  we  find  the 
contents  to  be  41     3'  10"  cubic  feet. 


2 

4' 

11 

8" 

3 

3' 

8" 

12 

6' 

39 

8' 

0 

1 

7' 

10" 

41      3'     10" 


2.  Multiply  9/7.  6'  by  4/if.  7'. 

3.  Multiply  12/^.  5'  by  Qft.  8'. 

4.  MuUiply  35/^.  4'  6"  by  9/?.  10'. 

5.  What  is  the  product  of  45//.  4'  3"  by  12^  2'  9"  ? 

6.  What  is  the  product  of  140//.  0'  2"  4'"  by  20//.  10'  ? 

7.  What  is  the  product  of  279//.  10'  6"  by  8'  4"  ? 

8.  What  are  the  contents  of  a  board  14//.  6'  3"  long  and 
2//.  9'  wide  ? 

9.  How  many  square  feet  in  a  floor  18//.  9'  long,  and  15/f. 
10'  wide  ? 

10.  How  many  square  yards  in  a  ceiling  70//.  9'  long,  and 
12//.  3'  wide  ? 


148  DnasioN  OF 

11.  What  will  be  the  cost  of  paving  a  yard  64/);.  6^  square, 
at  5  cents  a  square  foot  ? 

12.  What  are  the  cubic  contents  of  a  block  of  marble,  6/?.  9' 
long,  4/'/.  8'  wide,  and  2//!.  10'  thick  ? 

13.  There  is  a  room  97/7.  4'  around  it;  it  is  9//.  G'  high: 
what  will  it  cost  to  paint  the  walls,  at  18  cents  a  square  yard  ? 

14.  How  many  cubic  feet  of  wood  in  a  pile  36/^.  5'  long, 
6//.  8'  high,  and  3//.  6'  wide  ? 

15.  What  will  a  pile  of  wood  26/if.  8'  long,  6//'.  6'  high,  and 
Sft.  3'  wide,  cost,  at  $3,50  a  cord  ? 

16.  How  many  cubic  yards  of  earth  were  dug  from  a  cellar 
which  measured  38/V.  10'  long,  20ft.  6'  wide,  and  9/V.  4'  deep? 

17.  At  16  cents  a  yard,  what  will  it  cost  to  plaster  a  room 
22/;.  8'  long,  18/^  9'  wide,  and  lift.  6'  high?  There  are  to 
be  deducted  8  windows,  Gft.  4'  high  and  2/7.  9'  wide ;  2  doors, 
1ft.  6'  high  and  ^ft.  2'  wide,  and  the  base  moulding,  which  is 
1  foot  wide. 

DIVISION  OF  DUODECIMALS. 

142.  Division  of  Duodecimals  is  the  operation  of  finding 
from  two  duodecimal  numbers  a  third,  which  multiplied  by  the 
first,  will  give  the  second. 

1.  A  hall  contains  lOSsq.ft.  4'  5"  8'"  4^  and  is  Gft.  IV  S" 
wide  :  what  is  its  length  ? 

AnATA'SIS. The  OPF.RATION. 

units  of  the  dividend         ft.  s(].  ft.  ft. 

are  square  feet   and  6  11'  8")  103    4'  5"  8'"  4-(14  9'  11" 

fractions  of  a  square 
foot.  The  units  of 
the  divisor  arc  linear 
feet  and  fractions  of 
a  linear  foot. 

First,  consider  how 
often  the  first  two  parts  of  the  divisor  are  contained  in  the  first  part 
of  the  dividend.     The  first  two  parts  of  the  divisor  arc  nearly  equa^ 


97 

7' 

4" 

5 

9' 

1" 

8'" 

5 

2' 

9" 

0'" 

6' 

4" 

8"'  i' 

fi' 

4" 

8'"  4 

142.  What  is  Division  of  Duodecimals  1     How  is  it  performed  ] 


DIVISION    OF   DUODECIMALS.  149 

to  7  feet,  and  this  is  contained  in  I02sg.  ft.  14  times  and  something 
over.  Multiplying  the  divisor  by  this  term  of  the  quotient  and  sub- 
tracting, we  find  the  remainder  5ft.  9'  1",  to  which  bring  down  8"'. 

Next,  consider  how  many  times  the  first  two  parts  of  tlie  divisor, 
(equal  to  7  feet,  nearly.)  are  contained  in  the  first  two  parts  of  the 
remainder,  reduced  to  the  next  lower  unit ;  that  is,  5ft.  9'  =  69'. 
Multiplying  the  divisor  by  the  quotient  figure  9',  and  making  the 
subtraction,  we  have,  6'  4"  8",  to  which  bring  down  4'"'. 

Consider,  again,  how  often,  nearly  7  feet  is  contained  in  6'  4"=76" 
Multiplying  the  divi-sor  by  the  quotient  11",  we  find  a  product  equal 
to  the  last  remainder.     Hence,  the  process  of  division  is  the  same  as 
that  of  other  denominate  numbers^  except  in  the  manner  of  selecting  the 
quotient  figure. 

Notes. — 1.  If  the  integral  unit  of  the  dividend  and  divisor  is  the 
same,  the  unit  of  the  quotient  unll  he  abstract. 

2.  If  the  unit  of  the  dividend  is  a  superficial  unit,  and  the  unit  of 
the  divisor  a  linear  unit,  the  unit  of  the  quotient  will  be  linear. 

3.  If  the  unit  of  the  dividend  is  a  unit  of  volume,  and  the  unit  of 
the  divisor  linear,  the  unit  of  the  quotient  loill  be  superficial. 

4.  If  the  unit  of  the  dividend  is  a  unit  of  volume,  and  the  unit  of 
the  divisor  superficial,  the  unit  of  the  quotient  will  be  linear. 

EXAMPLES. 

1.  Divide  2^sq.fL  0'  4"  by  Qft.  4'. 

2.  Divide  bOsq.ft.  0^  10"  6'"  by  9/;.  6'. 

3.  What  is  the  length  of  a  floor  whose  area  is  1176sg'./i;.  1' 
6",  and  breadth  24/;.  3'? 

4.  A  load  of  wood,  containing  ll^cv.fL  2'  6"  8'",  is  Zft.  4' 
high,  and  4ft.  2'  wide  :  what  is  its  length  ? 

5.  In  a  granite  pillar  there  are  IQocu.ft.^  1"  6'";  it  is 
?>ft.  9'  wide,  and  1ft.  3'  thick  :  what  is  its  length  ? 

6.  There  are  394sg./i!.  2'9"in  the  floor  of  a  hall  that  is  \^ft. 
7'  wide  :  what  is  its  length  ? 

7.  A  board  Ylft.  6'  long,  contains  Tl  sq.ft.  8'  6"  :  what  is  its 
width  ? 

8.  From  a  cellar  42/V.  10'  long,  12/.  C  wide,  were  thrown 
158c2«.  yds.  lieu,  ft  4'  of  earth  :  how  deep  was  it? 


150  DECIMAL    FKACTIOrJS. 


DECIMAL    FRACTIONS. 

143.  There  are  two  kinds  of  Fractions  :  Common  Fractions 
and  Decimal  Fractions. 

A  Common  Fraction  is  one  in  which  the  unit  is  divided  into 
any  number  of  equal  parts. 

A  Decimal  Fraction  is  one  in  which  the  unit  is  divided 
according  to  the  scale  of  tens, 

144.  If  the  unit  1  be  divided  into  10  equal  parts,  each  part 
is  called  one-tenth. 

If  the  unit  1  be  divided  into  one  hundred  equal  parts,  each 
part  is  called  one-lmndrcdth. 

If  the  unit  1  be  divided  into  one  thousand  equal  parts,  the 
parts  are  called  thousandths,  and  we  have  like  expressions  for 
the  parts,  when  the  unit  is  further  divided  according  to  the 
scale  of  tens. 

These  fractions  may  be  written  thus  : 

Three-tenths,  ------  ^. 

Seventh-tenths,        -----  ^^. 

Sixty-five  hundredths,      -         -         -         -  _f^. 

215  thousandths,      -         -         .         -         - 
1275  ten  thousandths,       -         -         -         - 


2  1  5 
1000' 
12-5 

loooo- 


From  which  we  see,  that  in  each  case  the  denominator  indi- 
cates the  fractional  unit ;  that  is,  determines  whether  the  parts 
are  tenths,  hundredths,  thousandths,  &c. 


143.  How  many  kinds  of  fractions  are  tli-ere  ?  What  are  thoyi  What 
is  a  common  fractionl    What  is  a  flrcimal  fraction  1 

144.  When  the  unit  1  is  divided  into  10  equal  parts,  what  is  each  part 
called  1  What  is  each  pari  called  when  it  is  divided  into  100  equal  parts  ? 
When  into  1000  1  Into  10,000,  cVc.  1  How  are  decimal  fractions  formed? 
What  gives  denomination  to  the  fraction  1 


DECIMAL   FKACTIONS.  151 

145.  The  denominators  of  decimal  fractions  are  s(!ldom  set 
down.  The  fractions  are  usually  exj^ressed  by  means  of  a 
pei'iod,  placed  at  the  left  of  the  numerator. 

Thus,         y^  -         •       is  written     -         -       .3 

Too  -  -  -  -  .UO 

__2JL5_ .215 

1000  \  .i--!^" 

10000  .x-i^/ 

Tliis  method  of  writing  decimal  fractions  is  a  form  of  lan- 
guage, employed  to  avoid  writing  the  denominatoi's.  The  denomi- 
nator, however,  of  every  decimal  fraction  is  always  understood  : 

It  is  the  unit  1  with  as  many  ciphers  annexed  as  there  are 
places  of  figures  in  the  decimal. 

The  place  next  to  the  decimal  point,  is  called  the  place  of 
tenths,  and  its  unit  is  1  tenth.  The  next  place,  at  the  right,  is 
the  place  of  hundredths,  and  its  unit  is  1  hundredth  ;  the  next 
is  the  place  of  thousandths,  and  its  unit  is  1  thousandth  ;  and 
similarly  for  places  still  to  the  right. 

DECIMAL  NUMERATION  TABLE. 

w 

.^3  2 

•    tn    S    O  "S 

'"  ^    '^  ^  t^ 

_rf  T^    ™  '^  •    S 

•£  rg    tn   -^  ui    o 

m    i,    g  ^    iH    C    a 

c  =  o  c  -  =  c 

hKhhKSh 

.4  is  read  4  tenths. 

.5  4  -  -  54  hundredths. 

.0  6  4  -  -  64  thousandths. 

.6754  -  -  6754  ten  thousandths. 

.01234  -  -  1234  hundred  thousandths. 

.007654  -  -  7654  millionths. 

.0043604  -  -  43604  ten  millionths. 

Note. — Decimal  fractions  are  numerated  from  left  to  right ;  thus, 
tenths,  hundredths,  thousandths,  &c. 


152  DECIMAL   FRACTIONS. 

146.  Write  and  numerate  the  following  decimals  : 
Six-tenths,         -         -         -         -         .6 
Six  hundredths,  -         -         -         .0  G 

Six  thousandths,         -         -         -         .0  0  6 
Six  ten  thousandths,  -         -         -         .0006 
Six  hundred  thousandths,  -         -         .00006 
Six  miUionths,  -         -         -         -         .000006 
Six  ten  millionths,      -         -         -         .0000006 
Here  we  see  that  the  same  figure  denotes  different  decimal 
units,  according  to  the  place  which  it  occupies  ;  therefore, 

The  value  of  the  unit,  in  the  different  places,  in  passing  from 
the  left  to  the  right,  diminishes  according  to  the  scale  of  tens. 

Hence,  ten  of  the  units  in  any  place,  are  equal  to  one  unit  in 
the  place  next  to  the  left ;  that  is,  ten  thousandths  make  one 
hundredth,  ten  hundredths  make  one-tenth,  and  ten-tenths  make 
the  unit  1. 

This  scale  of  increase,  from  the  right  hand  towards  the  left,  is 
the  same  as  that  in  whole  numbers ;  therefore, 

Whole  numbers  and  decimal  fractions  may  be  united  by 
placing  the  decimal  point  between  them  ;  thus. 

Whole  numbers.  Decimals. 


to 

■5 

CO 

5 

w 

TS 

en 

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t/3 

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CA 

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3 

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4 

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4 

5 

1 

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0 

4 

3 

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7 

8. 

145.  Arc  the  denominators  of  decimal  fractions  generally  set  down  1 
How  are  Ihc  fractions  expressed  \  Is  the  denominator  understood  !  M'hat 
is  it  ?  What  is  the  place  ne.Yl  the  decimal  point  called  !  What  is  its 
unit!  What  is  the  next  place  called  ?  What  is  its  unitT  What  is  the 
third  place  called  !  What  is  its  unit  1  Which  way  are  decimals  numerated  ^ 


DECIMAL   FRACTIONS.  153 

A  number  composed  pai-tly  of  a  whole  number,  and  partly 
of  a  decimal,  is  called  a  mixed  number. 

RULE   FOR   WRITING   DECIMALS. 

Write  the  decimal  as  if  it  were  a  ichole  number,  prefixing  as 
many  ciphers  as  are  necessary  to  make  it  of  the  required  denomi- 
nation. 

RULE   FOR   READING   DECIMALS. 

Read  the  decimal  as  though  it  were  a  whole  number^  adding 
the  denomination  indicated  by  the  lowest  decimal  unit. 

EXAMPLES. 

Write  the  following  numbers  decimally : 

(1.)     (2.)     (3.)      (4.)      (5.) 

6  11  5  _2_7_  i±_ 

Too      10      1000      100      Tooo* 

(6.)      (7.)      (8.)      (9.)      (10.) 
"loo     'Tooo     *^ioo     ■^'^loo     -^''lO' 

"Write  the  following  numbers  in  figures,  and  numerate 
them : 

1.  Twenty-seven,  and  four-tenths. 

2.  Thirty-six,  and  fifteen  thousandths. 

3.  Ninety-nine,  and  twenty-seven  ten  thousandths. 

4.  Three  hundred  and  twenty  thousandths. 

5.  Two  hundred,  and  three  hundred  and  twenty  millionths. 

6.  Three  thousand  six  hundred  ten  thousandths. 

7.  Five, and  three  millionths. 

8.  Forty,  and  nine  ten  millionths. 


146.  On  what  does  the  unit  of  a  figure  depend  1  How  does  the  value 
change  from  the  left  towards  the  right  1  What  do  ten  units  of  any  ono 
place  make  1  How  do  the  units  of  the  places  increase  from  the  right  to- 
wards the  left  ^  How  may  whole  numbers  be  joined  with  decimals  !  What 
is  such  a  number  called  \  Give  the  rule  for  writing  decimal  fractions. 
Give  the  rule  for  reading  decimal  fractions. 


154  DECIMAL   FRACTIONS. 

9.  Forty-nine  hundred  ten  thousandths. 

10.  Fifty-nine  and  sixty-seven  ten  thousandths. 

11.  Four  hundred  and  sixty-nine  ten  thousandths. 

12.  Seventy-nine,  and  four  hundred  and  fifteen  millionths. 

13.  Sixty-seven,  and  two  hundred  and  twenty-seven  ten 
thousandths. 

14.  One  hundred  and  five,  and  ninety-five  ten  millionths. 

UNITED  STATES  MONEY. 

147.  The  denominations  of  United  States  Money  correspond 
to  the  decimal  division,  if  we  regard  1  dollar  as  the  unit. 

For,  the  dimes  are  tenths  of  the  dollar,  the  cents  are  hun- 
dredths of  the  dollar,  and  the  mills,  being  tenths  of  the  cent,  are 
thousandths  of  the  dollar. 

EXAMPLES. 

1.  Express  $37  and  26  cents  and  5  mills,  decimally. 

2.  Express  $17  and  5  mills,  decimally. 

3.  Express  $215  and  8  cents,  decimally. 

4.  Express  $275  5  mills,  decimally. 

5.  Express  $9  8  mills,  decimally. 

6.  Express  $15  6  cents  9  mills,  decimally. 

7.  Express  $27  18  cents  2  mills,  decimally. 

ANNEXING  AND  PREFIXING  CIPHERS. 

148.  Annexing  a  cipher  15  placing  it  on  the  right  of  a  number. 
If  a  cipher  is  annexed  to  a  decimal  it  makes  one  7nore  decimal 

place,  and,  therefore,  a  cipher  must  also  be  added  to  the  denomi- 
nator (Art.  145). 

The  numerator  and  denominator  will  therefore  have  been 
multiplied  by  the  same  number,  and  consequently  the  value  of 
the  fraction  will  not  be  changed  (Art.  118)  :  hence, 

147.  If  the  denominations  of  Federal  Money  he  expressed  decimally, 
■what  is  the  unit  1  What  part  of  a  dollar  is  1  dime  ?  What  part  of  a  dime 
is  a  centl  What  part  of  a  cent  ia  a  milH  What  part  of  a  dollar  is  I 
cent  1     1  mill  1 


DECIMAL   FRA.CTIOi!fS.  155 

Annexing  ciphers  to  a  decimal  fraction  does  not  alter  its  value. 
We  may  take  as  an  example,  .5  =  -f^. 

If  we  annex  a  cipher  to  the  numerator,  we  must,  at  the  same 
time,  annex  one  to  the  denominator,  which  gives, 

.5  =:     Y^     =  .50       by  annexing  one  cipher. 
.5  =    tVoV    —  '^0*^     ^y  annexing  two  ciphers, 
.5  =  yo%%'V   =  .5000  by  annexing  three  ciphers. 

Also,  .■*    _    y^j     _    .t\J    _    j-^Q     _    .'iW    —     ^000* 

Also,         .7  =  .70  =  .700  =  .7000  =  .70000. 

149.   Prefixing  a  cipher  is  placing  it  on  the  left  of  a  number. 

If  cij)hers  are  prefixed  to  the  numerator  of  a  decimal  frac- 
tion, the  same  number  of  ciphers  must  be  annexed  to  the  de- 
nominator. Now,  the  numerator  will  remain  unchanged  while 
the  denominator  will  be  increased  ten  times  for  every  cipher 
annexed ;  and  hence,  the  value  of  the  fraction  will  be  dimin- 
ished ten  times  for  every  cipher  prefixed  to  the  numerator 
(Art.  117). 

Prefixing  ciphers  to  a  decimal  fraction  diminishes  its  value 
ten  times  for  every  cipher  prefixed. 

Take,  for  example,  the  fraction   .3  =  ^^. 

.3  becomes      -^^      =  .03         by  prefixing  one  cipher. 

.3  becomes     jWo      =  .003       by  prefixing  two  ciphers. 

.3  becomes    yHqq     —  .0003     by  prefixing  three  ciphers: 
in  which  the  fraction  is  diminished  ten  times  for  every  cipher 
prefixed. 

148.  When  is  a  cipher  annexed  to  a  number?  Does  the  annexiinr  of 
ciphers  to  a  decimal  alter  its  value  1  Why  not  ?  What  does  five-tenths 
become  by  annexing  a  cipher  1  What  by  annexing  two  ciphers  !  Three 
ciphers  \  AVhat  does  7  tenths  become  by  annexing  a  cipher  ?  By  annex- 
ing two  ciphers  1     By  annexing  three  ciphers  1 

149.  When  is  a  cipher  prefixed  to  a  number  1  Whe»  prefixed  lo  a  decimal, 
does  it  increase  the  numerator  \  Does  it  increase  the  denominator  '\  What 
effect,  then,  has  it  on  the  value  ef  the  fraction  ? 


156 


ADDITION    OF 


ADDITION  OF  DECIMALS. 

150.  Addition  of  Decimals  is  the  operation  of  finding  the 
sum  of  twe  or  more  decimal  numbers. 

It  must  be  remembered,  that  only  units  of  the  same  value 
can  be  added  together.  Therefore,  in  setting  down  decimal 
numbers  for  addition,  figures  expressmg  the  same  unit  must  be 
placed  in  the  same  column. 

The  addition  of  decimals  is  then  made  in  the  same  manner 
as  that  of  whole  numbers. 

I.  Find  the  sum  of  87.06,  327.3  and  .0567. 

OPERATION. 

Place  the  decimal  points  in  the  same  column  :  87.06 

this  brings  units  of  the  same  value  in  the  same  327.3 

column  :  then  add  as  in  whole  numbers  j  hence,  .0567 

414.4167 

Rule. — I.    Set  doion   the  numbers   to  be 
added  so  that  figures  of  the  same  unit  value  shall  stand  in  the 
same  column. 

II.  Add  as  in  simple  numbers,  and  2'>oint  off  in  the  sum,  from 
the  right  hand,  a  number  of  places  for  decimals  equal  to  the 
greatest  number  of  places  in  any  of  the  numbers  added. 

Proof. — The  same  as  in  simple  numbers. 

EXAMPLES. 

1.  Add  6.035,  763.196,  445.3741,  and  91.5754  together. 

2.  Add  465.103113,  .78012,  1.34976,  .3549,  and  61.11. 

3.  Add  57.406  +  97.004  +  4  +  .6  +  .06  +  .3. 

4.  Add  .0009  +  1.0436  +  .4  +  .05  +  .047. 

ITjO.  What  is  Atldition  1  ^Vhat  parts  of  unity  may  be  added  together  1 
How  do  you  set  down  the  numbers  for  addition  !  How  will  the  decimal 
points  fain  How  do  you  then  add!  How  many  decimal  places  do  j-ou 
point  off  in  the  sum  1 


DECIMAL   FEACTIONS.  157 

5.  Add  .0049  +  49.0426  +  37.0410  +  360.0039. 

6.  Add  5.714,  3.456,  .543,  17.4957  together. 

7.  Add  3.754,  47.5,  .00857,  37.5  together. 

8.  Add  54.34,  .375,  14.795,  1.5  together. 

9.  Add  71.25,  1.749,  1759.5,  3.1  together. 

10.  Add  375.94,  5.732,  14.375,  1.5  together. 

11.  Add  .005,  .0057,  31.008,  .00594  together. 

12.  Required  the  sum  of  9  tens,  19  hundredths,  18  thou- 
sandths, 211  hundred-thousandths,  and  19  millionths. 

13.  Find  the  sum  of  two,  and  twenty-five  thousandths,  five, 
and  twenty-seven  ten-thousandths,  forty-seven,  and  one  hundred 
twenty  six-millionths,  one  hundred  fifty,  and  seventeen  ten-mil- 
lionths. 

14.  Find  the  sum  of  three  hundred  twenty-seven  thousandths, 
fifty-six  ten-thousandths,  four  hundred,  eighty-four  milHonths, 
and  one  thousand  five  hundred  sixty  hundred-millionths. 

15.  What  is  the  sum  of  5  hundredths,  27  thousandths,  476 
hundred-thousandths,  190  ten-thousandths,  and  1279  ten-mil- 
lionths  ? 

16.  "What  is  the  sum  of  25  dollars  12  cents  6  mills,  9  dollars 
8  cents,  12  dollars  7  dimes  4  cents,  18  dollars  5  dimes  8  mills, 
and  20  dollars  9  mills  ? 

17.  What  is  the  sum  of  126  dollars  9  dimes,  420  dollars 
75  cents  6  mills,  317  dollars  6  cents  1  mill,  and  200  dollars 
4  dimes  7  cents  3  mills  ? 

18.  A  man  bought  4  loads  of  hay,  the  first  contained  1  ton 
25  thousandths  ;  the  second,  997  thousandths  of  a  ton  ;  the  third, 
88  hundredths  of  a  ton ;  and  the  fourth,  9876  ten-thousandths 
of  a  ton  :  what  was  the  entire  weight  of  the  four  loads  ? 

19.  Paid  for  a  span  of  horses,  $225,50  ;  for  a  carriage? 
S127,055,  and  for  harness  and  robes,  $75,28  :  what  was  the 
entire  cost  ? 

20.  Bou'o-ht  a  barrel  of  flour  for  $9,375  ;  a  cord  of  wood  for 
$2,121 ;  a  barrel  of  apples  for  $1,621 ;  and  a  quarter  of  beef 
for  $6,09  :  what  was  the  amount  of  my  bill  ? 

21.  A  farmer  sold  grain,  as  follows :  wheat,  for  $296.75  ; 


158  SDBTK ACTION    OF 

corn,  for  $12G,12i;  oats,  for   $97,371;  rye,  for  $100,10  ;  and 
barley,  for  §oO,G2i :  what  was  the  amount  of  his  sale  ? 

22.  A  person  made  the  following  bill  at  a  store ;  5  yards  of 
cloth,  for  $16,408  ;  2  hats,  for  $4,87^ ;  4  pairs  of  shoes,  for  $6; 
20  yards  of  calico,  for  $2,  378  ;  and  12  skeins  of  silk,  for  $0,62^ : 
what  was  the  amount  of  his  bill  ? 


SUBTRACTION  OF  DECIMALS. 
151.  Subtraction  of  Dechials  is  the  operation  of  finding 
the  difference  between  two  decimal  numbers. 

I.  From  6.304  to  take  .0563. 

Note. — In  this  example  a  cipher  is  annexed  to        operation. 
the  minuend  to  make  the  number  of  decimal  places  6.3040 

equal  to  the  number  in  the  subtrahend.     This  docs  -05(33 

not  alter  the  value  of  the  mmuend  (Art.  148)  :  6.2477 

hence, 

KuLE. — 1.    Write  the  less  number  under  the  greater,  so  that 
figures  of  the  same  unit  value  shall  fall  in  the  same  column. 

II.  Subtract  as  in  simple  numbers,  and  point  off  the  decimal 
places  in  the  remainder,  as  in  addition. 

Proof. — Same  as  in  simi^le  numbers. 

EXAMPLES. 

1.  From  3278  take  .0879. 

2.  From  291.10001  take  41.496. 

3.  From  10.00001  take  .111111. 

4.  Required  the  difference  between  57.49  and  5.768. 

5.  What  is  the  difference  between  .3054  and  3.075  ? 

6.  Required  the  difference  between  1745.3  and  173.45. 

7.  AVhat  is  the  diflerence  between  seven-tenths  and  54  ten- 
thousandths  ? 


151.  Wliat  is  subtraction  of  decimal  fractions  !  How  do  you  set  down 
the  numbers  for  subtraction  1  How  do  you  then  subtract '  How  many 
Jcciinal  placcfl  do  you  point  off  \n  the  remainder  1 


DECIMAL    FRACTIONS.  159 

8.  What  is  the  difference  between  .105  and  1.00075  ? 

9.  What  is  the  difference  between  150.43  and  754.355  ? 

10.  From  1754.754  take  375.49478. 

11.  Take  75.304  from  175.01. 

12.  Required  the  difference  between  17.541  and  35.49. 

13.  Ecquired  the  difference  between  7  tenths  and  7  mil- 
lionths. 

14.  From  396  take  67  and  8  ten-thousandths. 

15.  From  1  take  one-thousandth. 

16.  From  6374  take  fifty-nine  and  one-tenth. 

17.  From  365.0075  take  5  millionths. 

18.  From  21.004  take  98  ten-thousandths. 

19.  From  260.3609  take  47  ten-millionths. 

20.  From  10.0302  take  19  millionths. 

21.  From  2.03  take  6  ten-thousandths. 

22.  From  one  thousand,  take  one-thousandth. 

23.  From  twenty-five  hundred,  take  twenty  five  hundredths. 

24.  From  two  hundred,  and  twenty  seven  thousandths,  take 
ninety-seven,  and  one  hundred  twenty  ten-thousandths. 

25.  A  man  owning  a  vessel,  sold  five  thousand  seven  hundred 
sixty-eight  ten  thousandths  of  her  :  how  much  had  he  left  ? 

26.  A  farmer  bought  at  one  time  127.25  acres  of  land,  at 
another,  84.125  acres,  at  another,  116.7  acres.  He  wishes  to 
make  his  farm  amount  to  500  acres  :  how  much  more  must  he 
purchase  ? 

27.  Bought  a  quantity  of  lumber  for  $617.37^,  and  sold  it 
for  $700  :  how  much  did  I  gain  by  the  sale  ? 

28.  Having  bought  some  cattle  for  §325.50 ;  some  sheep  for 
$97.12^;  and  some  hogs  for  $60.87i ;  I  sell  the  whole  for 
$510.10  :  what  was  my  entire  gain  ? 

29.  A  dealer  in  coal  bought  225.025  tons  of  coal ;  he  sold  to 
A,  1.05  tons,  to  B,  20.007  tons,  to  G,  40.1255  tons,  and  to 
D,  37.00056  tons  :  how  much  had  he  left  ? 

30.  A  man  owes  $2346.865,  and  has  due  him,  from  Ay 
$1240.06,  and  from  B,  $1867.981 :  how  much  will  he  have  left 
after  paying  his  debts  ? 


160  MULTIPLICATION    OF 

31.  Bought  of  eacli  of  two  persons,  1284.0-'5  pounds  of  wool, 
from  which  I  sell  to  three  persons,  each  262.125  pounds :  how 
much  will  I  still  have  on  hand  ? 


MULTIPLICATION  OF  DECIMAL  FRACTIONS. 

152.  Multiplication  of  decimal  fractions  is  the  operation 
of  taking  one  number  as  many  times  as  there  are  units  in  ano- 
ther, when  one  of  the  factors  contains  a  decimal,  or  when  they 
both  contain  decimals. 

1.  Multiply  8.03  by  G.102. 

OPERATION. 

Analysis. — If  we  change  both  factors 
to  common  fractions,  the  product  of  the 
mimerators  will  be  the  same  as  that  of 
the  decimal  numbers,  and  the  number  of 
decimal  places  loill  he  equal  to  the  number 
of  ciphers  in  the  two  denominators  ;  hence, 

Rule. — Multiply  as  in  simjyle  man-  48.99906 

hers,  andiwint  off  in  the  product,  from 

the  right  hand,  as  many  firjures  for  decimals  as  there  are  decimal 
places  in  both  factors  ;  and  if  there  be  not  so  many  in  the  pro- 
duct, supply  the  deficiency  by  prefixing  ci2}hc7-s. 

examples. 

1.  Muhiply  2.125  by  375  thousandths. 

2.  Muhiply  .4712  by  5  and  G  tenths. 

3.  INIuhiply  .0125  by  4  thousandths. 

4.  Muhiply  G.002  by  25  hundredths. 

5.  Muhiply  473.54  by  57  thousandths. 

G.  IMultiply  137.549  by  75  and  437  ^hou-sandths. 
7.  ]\Iultiply  3  and  .7495  by  73487. 

152.  After  multiplying,  how  many  decimal  places  will  you  point  oil'  :n 
the  product  1  When  tlicrc  are  not  so  many  in  tlie  product,  whjit  do  you 
do  1     Give  the  rule  for  the  multiplication  of  decimals. 


803 
100 

=  8.03 

6102 
1000 

=  6.102 
1606 

803 

4818 

DECIMAL   FRACTIONS.  161 

8.  Multiply  ,04375  by  4713 i  liundi-ed  thousandths. 

9.  Multiply  .371343  by  seventy-five  thousand  493. 

10.  Multiply  49.0754  by  3  and  5714  ten  thousandths. 

11.  Multiply  .573005  by  754  millionths. 

12.  Multiply  .375494  by  574  and  375  hundredths. 

13.  Multiply  two  hundred  and  ninety-four  millionths,  by  ono 
millionth. 

14.  Multiply  three  hundred,  and  twenty-seven  hundredths 
by  G2. 

15.  Multiply  93.01401  by  10.03962. 
IG.  Multiply  59G.04  by  0.000012. 

17.  Multiply  38049.079  by  0.000016. 

18.  Multiply  1192.08  by  0.000024. 

19.  Muhiply  7G098.158  by  0.000032. 

20.  Multiply  thirty-six  thousand,  by  thirty-six  thousandths. 

21.  Multiply  125  thousand,  by  25  ten  thousandths. 

22.  What  is  the  product  of  50  thousand,  by  75  ten  millionths  ? 

23.  What  is  the  product  of  48  hundredths,  by  75  ten  thou- 
sandths ? 

24.  What  are  the  contents  of  a  lot  of  land  16.25  i-ods  long, 
and  9.125  rods  wide  ? 

25.  What  are  the  contents  of  a  board  12.07  feet  long,  and 
1.005  feet  wide? 

26.  What  will  27.5  yai'ds  of  cloth  cost,  at  .875  dollars  per  yard  ? 

27.  At  $25,125  an  acre,  what  will  127.045  acres  cost? 

28.  Bought  17.875  tons  of  hay,  at  $11.75  a  ton:  what  was 
the  cost  of  the  whole  ? 

29.  A  gentleman  purchased  a  farm  of  420.25  acres,  for 
$35.08  an  acre  ;  he  afterwards  sold  196.175  acres  to  one  man 
for  $37.50  an  acre,  and  the  remainder  to  another  person,  for 
$36,125  an  acre  :  what  did  he  gain  on  the  first  cost  ? 

30.  A  merchant  bought  two  pieces  of  cloth,  one  containing 

87.5  yards,  at  $2.75  a  yard,  and  the  other,  containing   27.35 

yards,  at  $3,125  a  yard ;  he  sold  the  whole  at  an  average  price 

of  $2.94  a  yard  :  did  he  gain  or  lose  by  the  bargain,  and  how 

much  ? 

8 


162  CONTKACTIOXS    IN 

CONTRACTIONS  IN  MULTIPLICATION. 

153.  Contraction,  in  the  multiplication  of  decimals,  is  a 
short  method  of  finding  the  product  of  two  decimal  numbers  in 
such  a  manner,  that  it  shall  contain  but  a  given  number  of 
decimal  places. 

1.  Let  it  be  required  to  find  the  product  of  2,38645  multi- 
plied by  38.2175,  in  such  a  manner  that  it  shall  contain  but  four 
decimal  places. 

Analysis. — It  is  proposed,  in  this  example,  to  operation. 

take  the  multiplicand  2.38645,   38  times,  then  2.38645 

2    tenths  times,  then  1   hundredth  times,  then  5712.83 

7    thousandths  times,    then    5    ten-thousandths  715935 

times,   and  the  sum  of  these  several  products  190916 

will  be  the  product  sought.  4773 

Write  the  unit  figure  of  the  multiplier  directly  239 

under  that  place  of  the  multiplicand  which  is  167 

to  be  retained  in  the  product,  and  the  remaining  1 2 

places  of  integer  figures,  if   any,  at  the  right,  91.2042 
and  then  write  the  decimal  places  at  the  left  in 
their  order,  tenths,  hundredths,  &c. 

When  the  numbers  are  so  written,  the  product  of  any  figure 
in  the  multiplier  hij  the  figure  of  the  multiplicand  directly  over 
it,  will  be  of  the  same  order  of  value  as  the  last  figure  to  be  re- 
tained in  the  product. 

Therefore,  the  first  figure  of  each  product  is  always  to  be 
arranged  directly  under  the  last  retained  figure  of  the  multlpli- 
cand.  But  it  is  the  whole  of  the  multiplicand  which  should  be 
multiplied  by  each  figure  of  the  multiplier,  and  not  a  j^art  of  it 
only.     Hence,  to  compensate  for  the  part  omitted,  we  begin 

153.  What  is  contraction  in  the  multiplication  of  decimals  ?  What  is 
proposed  in  the  example  !  liow  arc  the  numbers  written  down  for  multi- 
(jlication  1  When  the  numbers  are  so  written,  what  will  be  the  oriicr  of 
value  of  the  product  of  any  figure  of  the  multiiilier  by  the  figure  directly 
over  it  1  Where  then  is  the  first  figure  of  each  product  to  be  written ! 
liow  do  you  compciibutc  for  tlie  j»art  ouiiUvd  ! 


\ 


MULTIPLICATION.  163 

with  tlie  figure  at  tlie  riglit  of  the  one  directly  over  any 
muUiplier,  and  carry  one  when  the  product  is  greater  than 
5  and  less  than  15,  2  when  it  falls  between  15  and  25,  3 
when  it  falls  between  25  and  35,  and  so  on  for  the  higher 
numbers. 

For  example,  when  we  multiply  by  the  8,  instead  of  saying 
8  times  4  are  32,  and  writing  down  the  2,  we  say  first,  8  times 
5  are  40,  and  then  carry  4  to  the  product  32,  which  gives  36. 
So,  when  Ave  multiply  by  the  last  figure  5,  we  first  say,  5  times 
3  are  15,  then  5  times  2  are  10  and  2  to  carry  make  12,  which 
is  written  down 

EXAMPLES. 

1.  Multiply  36.74637  by  127.0463,  retaining  three  decimal 
places  in  the  product. 

Contraction.  Oonunon  ivay. 

36.74637  36.74637 

3640.721  127.0463 


3674637  11023911 

734927  22047822 

257225  14698548 

1470  25722459 

220  7349274 

11  3674637 


4668.490  4668.490346931 

2.  Multiply  54.7494367  by  4.714753,  reserving  five  places 
of  decimals  in  the  product. 

3.  Multiply  475.710564  by  .3416494,  retaining  three  decimal 
places  in  the  product. 

4.  Multiply  3754.4078  by  .734576,  retaining  five  decimal 
places  in  the  product. 

5.  Multiply  4745.679  by  751.4549,  and  reserve  only  whole 
numbers  in  the  product. 


16-i 


DIVISION    OF 


154.  Note. — When  a  decimal  num'ber  is  to  be  multiplied  by  10, 
100,  1000,  kc,  the  multiplication  may  be  made  by  removing  the 
decimal  point  as  many  places  to  the  right  hand  as  there  are  ciphers 
in  the  multiplier ;  and  if  there  be  not  so  many  figures  on  the  right 
of  the  decimal  point,  supply  the  deficiency  by  annexing  ciphers. 


Thus,  4.27  multiplied  by 


Also,  59G.027  multiplied  by 


10 

42.7 

100 

427 

>  1000 

>  =  < 

4270 

10000 

42700 

100000 

^  427000 

10    ' 

'    5960.27 

100 

59G02.7 

>  1000 

-  =  ^ 

596027 

10000 

5960270 

100000 

59602700 

I 


i 


DIVISIOxN  OF  DECIMAL  FRACTIONS. 
1.5.5.  Division  of  Decim.^l   Fractions   i.s  the   operation 
of  divi.sion  when  eitlier  the  divisor  or  dividend  is  a  decimal,  or 
when  both  are  decimals. 


Analysis. — Since  the  dividend  must  be 
equal  to  the  product  of  the  divisor  and  quo- 
tient, it  must  contain  as  many  decimal 
places  as  both  of  them  (Art.  152)  :  there- 
fore. 

There  must  be  as  many  decimal  places  in  the 
quotient  as  the  number  of  decimal  places  in  the 
dividend  exceeds  thnt  in  the  divisor  :  hence, 


OPERATION. 

2.043). 71505(35 
^129 
10215 
10215 


Ans.    0.35. 


Rule. — Divide  as  in  simple  numhers,  and  point  off  in  the 


154.  How  do  you  multiply  a  decimal  number  by  10,  100.  1000,  &c.  ? 
If  there  are  not  as  many  decimal  figures  as  there  are  ciphers  in  the  nnilti- 
pUer,  what  do  you  do  ? 

155.  If  one  decimal  fraction  be  multiplied  by  anothor,  how  many  deci- 
mal places  will  there  be  in  the  product '  How  docs  the  number  of  deci- 
mnl  jilaces  in  the  dividend  compare  with  those  in  the  divisor  and  quotient' 
How  do  you  determine  the  number  of  decimal  places  in  the  quotient  1 
Clivc  the  rule  for  the  division  of  decimals. 


DECIMAL   FE ACTIONS.  165 

quotie7it,from  the  right  hand,  as  mamj  -places  for  decimals  as  the 
numhcr  of  decimal  'places  in  the  dividend  exceeds  that  in  the  divi- 
sor ;  and  if  there  are  not  so  many,  mpphj  the  deficiency  hy  pre- 
fixing ciphers. 


EXAMPLES. 


1.  Divide  4.6842  by  2.11. 

2.  Divide  12.825G1  by  1.505. 

3.  Divide  33.6G431  by  1.01. 

4.  Divide  .010001  by  .01. 

5.  Divide  24.8410  by  .002. 

6.  Divide  .0125  by  2.5. 


7.  Divide  .051  by  .012. 

8.  Divide  .0G3  by  9. 

9.  Divide  1.05  by  14. 

10.  Divide  5.1435  by  4.05. 

11.  Divide  .46575  by  31.05. 

12.  Divide  2.46616  by  .145. 


13.  Wliat  is  the  quotient  of  75.15204,  divided  by  3  ?     By  .3  ? 
By  .03  ?     By  .003  ?     By  .0003  ? 

14.  What   is  the    quotient  of  389.27688,    divided    by    8? 
By  .08  ?     By  .008  ?     By  .0008  ?     By  .00008  ? 

15.  What  is  the  quotient  of  374.598,  divided  by  9  ?     By  .9  ? 
By  .09  ?     By  .009  ?     By  .0009  ?     By  .00009  ? 

16.  What  is  the  quotient  of  1528.4086488,  divided  by  6? 
By  .06  ?     By  .006  ?     By  .0006  ?     By  .00006  .?    By  .000006  ? 

17.  Divide  17.543275  by  125.7. 

18.  Divide  1437.5435  by  .7493. 

19.  Divide  .000177089  by  .0374. 

20.  Divide  1674.35520  by  9.60?, 

21.  Divide  120463.2000  by  1728. 

22.  Divide  47.54936  by  34.75. 

23.  Divide  74.35716  by  .00573. 

24.  Divide  .37545987  by  75.714. 

25.  If  25  men  remove  154,125  cubic  yards  of  eartli  in  a  day, 
how  much  does  each  man  remove  ? 

26.  If  167  dollars  8  dimes  7  cents  and  5  mills  be  equally 
divided  among  17  men,  how  much  will  each  receive  ? 

27.  Bought  45.22  yards  of  cloth  for  ^97.223  :  how  much  was 
it  a  yard  ? 

28.  If  375.25  bushels  of  salt  cost  $232,655,  wliat  is  the  price 
per  bushel  ? 


L66  DIVISION    OF 

29.  At  $0,125  per  pound,  how  much  sugar  can  be  bought  for 
$2.25  ? 

30.  How  many  suits  of  clothes  can  be  made  from  34  yards 
of  cloth,  allowing  4.25  yards  for  each  suit  ? 

31.  If  a  man  travel  26.18  miles  a  day,  how  long  will  it  take 
him  to  travel  366.52  miles? 

32.  A  miller  wishes  to  purchase  an  equal  quantity  of  wheat, 
corn,  and  rye ;  he  pays  for  the  wheat,  $2,225  a  bushel ;  for  the 
corn,  $0,985  a  bushel ;  and  for  the  rye,  SI. 168  a  bushel:  how 
many  bushels  of  each  can  he  buy  for  S242.979  ? 

33.  A  farmer  purchased  a  farm  containing  56  acres  of  wood- 
land, for  which  he  paid  $46,347  per  acre  ;  176  acres  of  meadow 
land,  at  the  rate  of  $59,465  per  acre  ;  besides  which  there  was 
a  swamj)  on  the  farm  that  covered  37  acres,  for  which  he  was 
charged  $13,836  per  acre.  What  was  the  area  of  the  land  ; 
what  its  cost ;  and  what  the  average  price  per  acre  ? 

34.  A  person  dying  has  $8345  in  cash,  and  6  houses,  valued 
at  $4379.837  each  ;  he  ordered  his  debts  to  be  i)aid,  amounting 
to  §3976.480,  and  $120  to  be  expended  at  his  funeral ;  the 
residue  was  to  be  divided  among  his  five  sons  in  the  following 
manner :  the  eldest  was  to  have  a  fourth  part,  and  each  of  the 
other  sons  to  have  equal  shares.  What  was  the  share  of  each 
son  ? 


( 


PARTICULAR   CASES. 

156.  Note. — 1 .  When  any  decimal  number  is  to  be  divided  by  10, 
100,  1000,  &c.,  the  division  is  made  by  removing  the  decimal  point 
as  many  places  to  the  left  as  tlieie  are  O's  in  the  divisor  :  and  if 
there  be  not  so  many  figures  on  the  left  of  the  decimal  point,  the 
deficiency  must  be  supplied  by  prefixing  ciphers. 


l.'jG.  How  do  you  divide  a  decimal  number  by  10.  100,  1000,  &c.  1  II 
thorn  be  not  as  many  figures  at  tlie  left  of  tlic  deci.nal  point  as  there  cipher* 
in  the  divisor,  what  do  you  do  ? 


DECIMAL    FRACTIONS. 


167 


10          ] 

[4.987 

100 

.4'J87 

1000 

>  =  -^ 

.01987 

10000    J 

.004987 

10          1 

r    32.756 

100 

3.2756 

1000 

y  = 

[       .32756 

10000 

.032756 

100000 

.0032756 

49.87  divided  by 


327.56  divided  by 


157.  Note. — 2.  When  there  are  more  decimal  places  in  the  divi- 
sor than  in  the  dividend,  annex  as  many  ciphers  to  the  dividend  aa 
are  necessary  to  make  its  decimal  places  equal  to  those  of  the  dii^i 
sor;  all  the  figures  of  the  quotient  will  then  he  lohole  numbers.  Always 
bear  in  mind  that  the  quotient  is  as  many  times  greater  than  1,  as 
the  dividend  is  times  greater  than  the  divisor. 


EXAMPLES. 

1.  Divide  4397.4  by  3.49. 

We  annex  one  0  to  the  dividend.  Had 
it  contained  no  decimal  place,  we  should 
have  annexed  two. 


OPERATION. 

3.49)4397.40(1260 
349 


907 
698 
2094 
2094 


2.  Divide  1097.01097  by  .100001.  Ans.  1260 

3.  Divide  9811.0047  by  .1629735. 

4.  Divide  .1  by  one  ten-thousandths. 

5.  Divide  10  by  one-tenth. 

6.  Divide  6  by  .6.  By  .06.  By  .006.  By  .2.  By  .3.  By 
.003.     By  .5.     By  .005.     By  .000012. 

158«  Note. — 3.  When  it  is  necessary  to  continue  the  division 
farther  than  the  figures  of  the  dividend  will  allow,  we  may  annex 
ciphers  to  it,  and  consider  them  as  decimal  places. 

157.  If  there  are  more  decimal  places  in  the  divisor  than  in  the  dividend, 
what  do  you  do  1     AVhat  will  the  figures  of  the  quotient  then  be  1 

158.  How  do  you  continue  the  division  after  you  have  brought  down  all 
the  figures  of  the  dividend  ]  When  the  division  does  not  terminate,  what 
sign  do  you  place  after  the  quotient  1     What  does  it  show  1 


1G8 


CONTRACTIONS   IN 


EXAMPLES. 

1.  Divide  4.25  by  1.25. 

In  this  example,  after  having  exhausted 
the  decimals  of  the  dividend,  we  annex  an 
0,  and  then  the  decimal  places  vised  in  the 
dividend  will  exceed  those  in  the  divisor 
by  1. 


OPERATION. 

1.25)4.25(3.4 
3.75 
500 
500 


Ans.     3.4 


2.  Divide  .2  by  .06. 

We  see,  in  this  example,  that  the  division 
will  never  terminate.  In  such  cases  the  di- 
vision should  be  carried  to  the  third  or  fourth 
place,  which  will  give  the  answer  true 
enough  for  all  practical  purposes,  and  the 
sign  +  should  then  be  written,  to  show  that 
the  division  may  still  be  continued. 


OPERATION. 

.06). 20(3. 333  + 
18 

18 
20 

18 
20 


Ans.  3.333  + 


3.  Divide  37.4  by  4.5. 

4.  Divide  58G.4  by  375. 


5.  Divide  94.0360  by  81.032. 

6.  Divide  36.2678  by  2  25. 


159.  Note. — 4.  If  we  regard  1  dollar  as  the  unit  of  United  States 
Currency,  all  the  lower  denominations,  dimes,  cents,  and  mills,  are 
decimals  of  tiie  dollar.  Hence,  all  the  operations  uponUiiitid  States 
Money  arc  the  same  as  the  correspondins  operations  on  decimal 
fractious. 


CONTR.ICTIONS   IN   DIVISION. 

160.  Contractions  in  division  of  decimals,  are  short 
methods  of  finding  a  quotient  Avhich  shall  contain  a  given 
number  of  decimal  places. 


159.  M'liat  is  tlie  unit  of  the  currnicy  of  the  United  States  ?  '^^'llat 
parts  of  lhi.s  unit  arc  tlie  inferior  dcnoniinalions.  dimes,  cnits,  and  mills  1 

IGO.  "What  are  the  contractions  in  division?  E.\plaiii  the  process  of 
making  tho  ilivision  T 


DIVISION. 


EXAMPLES. 


169 


1.  Divide   754.347385   by    G1.34775,  and  find   a  quotient 
which  shall  contain  three  places  of  decimals. 


Common  Method. 
01.34775)754.34738500(12.296  Contracted  Method. 

75  61.34775)754.347385(12.290 
988  61348 


61347 


14086 
12260 


1817 
1226 

~590 
552 

"38 


550  14086 

438^  12269 

9550  1817 

48350  12^1 

12975  590 

353750  ^ 


361808650  38 

37 


1545100 

1 


In  the  operation,  by  the  common  method,  the  figures  at  the 
right  of  the   vertical  line,  do  not  aflect  the  quotient  figures  : 

1.  Note  the  unit  of  the  first  quotient  figure  and  then  note  the 
number  of  figures  which  the  quotient  is  to  contain. 

2.  Select  as  many  figures  of  the  divisor  as  you  wish  places  of 
figures  in  the  quotient,  and  midtiphj  the  figures  so  selected  by  the 
first  quotient  figure,  observing  to  carry  for  the  figures  cast  off  as 
in  the  contraction  of  multiplication. 

3.  Use  each  remainder  as  a  new  dividend,  and  in  each  follow 
ing  division  omit  one  figure  at  the  right  of  the  divisor. 

]S^OTE. — In  the  example  above,  the  order  of  the  first  quotient  figure 
M'ns  obviously  tens  :  hence  there  were  two  places  of  whole  numbers  ; 
and  as  there  were  three  decimal  places  required  in  the  quotient. /I'c 
fip:u)es  of  the  divisor  must  be  used. 

2.  Divide  59  by  .74571345,  and  let  the  quotient  contain  four 
pl.'u't's  of  decimals. 

160    What  fifrures  may  be  omitted  in  the  contracted  method'! 


ITO  KEDUCTION    OF 

3.  Divide    17493.407704962    by   495.783269,   and   let   the 
quotient  contain  four  places  of  decimals. 

4.  Divide   98.187437   by   8.4765G18,  and  let   the    quotient 
contain  seven  places  of  decimals. 

0.  Divide  47194.379457  by  14.73495,  and  let  the  quotient 
contain  as  many  decimal  places  as  there  will  be  integers  in  it. 

REDUCTION   OF    COMMON  AND  DECIMAL   FRACTIONS. 

161.    To  change  a  common  to  a  decimal  fraction. 

The  value  of  a  fraction  is  the  quotient  of  the  numerator  di 
vided  by  the  denominator  (Art.  105.) 

1.  Reduce  |-  to  a  decimal. 

Analysis. — If  we  place  a  decimal  point  after  the  7,     operation. 
and  then  write  any  number  of  O's,  alter  it,  the  value         8)7.000 
of  the  numerator  will  not  be  changed  (Art.  148).  .875 

If  then,  we  divide  by  the  denominator,  the  quo 
tient  will  be  the  decimal  number  :  hence, 

Rule. — Annex  decimal  ciphers   to  the  numerator  and  then 
divide  by  the  denominator,  pointing  off  as  in  divif^ion  of  decimal' 

EXAMPLES. 

Reduce  the  following  common  fractions  to  decimals : 
1.  Reduce  \,  l,  and  f.  9.  Reduce  j"^  and  ^^. 


i 


2.  Reduce  |^,  ^,  and  -f^. 

3.  Reduce  f  and  -^^. 

4.  Reduce  j^-^  and  y*j. 

5.  Reduce  l  and  tttoTT' 

6.  Reduce  -^^  and  \^. 

7.  Express  |i5^|  decimally. 

8.  Express  -^^^.j  decimally. 


10.  Express  3^}  ^5-  deciraullj 

11.  Reduce  y^j  and  2^/j." 

12.  Reduce  -^-  of  j  of  6. 

13.  Reduce  |  of  }\. 

14.  Reduce  y\  of  ||. 

15.  Reduce  |  of  r!-- 

16.  Reduce  ?°hind  ^*\. 

20  7  5  8 


17.  What  is  the  decimal  value  off  of  |-  multiplied  by  yg. 

161,   How  do  you  cliaiigcaconimon  toadccimal  fraction'! 

SfZ.   How  ilo  yoii  rh.iiii;'^  a  dcciiiial  to  llic  loriii  of  a  comiDoii  fraction? 


COMMON    FK ACTIONS.  171 

18.  What  is  the  value,  in  decimals,  of  -}  of  f  of  I  divided 
Djfoff? 

19  A  man  owns  |-  of  a  ship  ;  he  sells  2%  of  his  share  :  what 
part  is  that  of  the  whole,  expressed  in  decimals  ? 

20.  Bought  11  of  87fV  bushels  of  wheat  for  ^^^  of  7  dollars 
a  bushel :  how  much  did  it  come  to,  expressed  in  decimals  ? 

21.  If  a  man  receives  f  of  a  dollar  atone  time,  71  at  another, 
and  8|-  at  a  third :  how  much  in  all,  expressed  in  decimals  ? 

22.  What  mixed  decimal  is  equal  to  f  of  18,  and  /^  of  11 
plus  74,  added  together  ? 

23.  What  decimal  is  equal  to  f  of  31  taken  from  f  of  8f  ? 

24.  What  decimal  is  equal  to  4fj  V^j  h  added  together? 

162.    To  change  a  decimal  to  the  form  of  a  common  fraction. 
A  decimal  fraction  may  be  changed  to  the  form  of  a  common 
fraction  by  simj^ly  writing  its  denominator  (Art.  145). 

EXAMPLES. 

Express  the  following  decimals  in  vulgar  fractions. 


1.  Reduce  .25  and  .75. 

2.  Reduce  .125  and  .625. 

3.  Reduce  .105  and  .0025. 

4.  Reduce  .8015  and  .6042. 


5.  Reduce  .68375. 

6.  Reduce  .01875. 

7.  Reduce  .22575. 

8.  Reduce  .265625. 


DENOMINATE  DECIMALS. 

163.  A  Denominate  Decimal  is  one  in  which  the  unit  of 
the  fraction  is  a  denominate  number.  Thus,  .3  of  a  dollar,  .7  of 
a  shilling,  .8  of  a  yard,  &;c.,  are  denominate  decimals,  in  which 
the  units  are,  1  dollar,  1  shilling,  1  yard. 

CASE  I.  «• 

164.  To  find  the  value  of  a  denominate  number  in  decimals  of 
a  liigher  tmit. 


163.  What  is  a  denominate  {lecimaH 

164.  How  do  you  find  the  value  of  a  denominate  number  in  decimals  of 
a  higher  unit  I 


OPEUATION. 

■|f?.  =  .75d.  ;  hence, 

9f(/.  =9.75d. 

12)9.75(7. 

.8125s.,  and 

20)4.8125.5. 

172  REDUCTION    OF 

I.  Reduce  £1  4s.  9^d.  to  the  decimal  of  a  £. 

Analysis. — Wc  first  reduce  3 
fartliings  to  the  decimal  of  a  penny, 
by  dividing  by  4.  ^Ye  then  annex 
the  quotient  .756?.  to  the  9  pence. 
We  next  divide  by  12,  giving  .81 25, 
which  is  the  decimal  of  a  shilling. 

This  we  annex  to  the  shillings,  and  £.240625  :  therefore, 

then  divide  by  20.  ^1   4^   93J.  ='^^1.240625. 

Rule. — I.  Divide  the  lowest  denomination  hy  the  units  of  the 
scale  which  connect  it  toith  the  next  higher,  annexing  ciphers,  if 
nccessa?y. 

II.  Annex  the  quotient  to  the  next  higher  dcnominntion  and 
divide  hy  the  units  of  the  scale  ;  and  proceed  in  the  same  manner 
through  all  the  denominations,  to  the  required  unit. 

Note. — When  any  denomination,  between  the  highest  and  the 
lowest  is  wanting,  the  number  to  be  prefixed  to  the  corresponding 
quotient,  is  0. 

EXAMPLES. 

1.  Reduce  14  drams  to  the  decimal  of  a  Ih.,  Avoirdupois. 

2.  Reduce  78(f.  to  the  decimal  of  a  £. 

o.  Reduce  03  pints  to  the  decimal  of  a  peck. 

4.  Reduce  9  hours  to  the  decimal  of  a  day. 

5.  Reduce  375G78  feet  to  the  decimal  of  a  mile. 

6.  Reduce  7oz.  Iddict.  of  silver  to  the  decimal  of  a  pound, 

7.  Reduce  Sctct.  lib.  8oz.  to  the  decimal  of  a  ton. 

8.  Reduce  2.45  sliillings  to  the  decimal  of  a  £.  , 

9.  Reduce  1.047  roods  to  the  decimal  of  an  acre. 

10.  Reduce  176.9  yards  to  the  decimal  of  a  mile. 

11.  Reduce  2qr.  lilb.  to  the  decimal  of  a  C7rt. 

12.  Reduce  IO02;.  ISdivt.  ^('■•gr.  to  the  decimal  of  a  lb. 

13.  Reduce  oqr.  2na.  to  the  decimal  of  a  yard. 

14.  Reduce  ^gal.  to  the  decimal  of  a  hog.-^head. 

15.  Reduce  17//.  Gm.  43ser.  to  the  decimal  of  a  day. 

16.  Reduce  4ctct.  '^^qr.  to  the  decimal  of  a  ton. 

17.  Reduce  19s.  5d.  2fa7\  to  the  decimal  of  a  poiuid. 


DKNOMINATE   DECIMALS.  173 

18.  Reduce  1^.  ."7/'.  to  the  decimal  of  an  acre. 

19.  Reduce  2qr.  ona.  to  the  decimal  of  an  Eng.  Ell. 

20.  Reduce  2ijd.  2ft.  G^««.  to  the  decimal  of  a  mile. 

21.  Reduce  15'  22^."  to  the  decimal  of  a  degree. 

22.  Reduce  290  cubic  inches  to  the  decimal  of  a  ton  of  round 
timber. 

23.  Reduce  obusJi.  oph.  to  the  decimal  of  a  chaldron. 

24.  Reduce  17yc?.  1ft.  Qin.  to  the  decimal  of  a  mile. 

25.  "What  decimal  part  of  a  year  is  9^  months  ? 

2G.  What  decimal  part  of  a  lb.  is  lOoz.  18dwt.  16gr.? 

27.  What  decimal  part  of  an  acre  is  1^.  14P.  ? 

28.  AYhat  decimal  part  of  a  chaldron  is  Aojyk.  ? 

29.  What  decimal  part  of  a  mile  is  72  yards  ? 

30.  What  part  of  a  ream  of  paper  is  9  sheets  ? 

31.  What  part  of  a  rod   in  length  is  4.0125  inches? 

32.  Reduce  IQwk.  2£?a.  to  the  decimal  of  a  leap  year. 

33.  Reduce  4  5    13   19  lO^-r.  to  the  decimal  of  a  ife. 

34.  Reduce  3qt.  1.7 opt.  to  the  decimal  of  a  /ihd. 

35.  Reduce  24:sq.  yd.  l.^sq.ft.  to  the  decimal  of  an  acre. 

36.  Reduce  2qr.  Ina.  0.3 6m.  to  the  decimal  of  a  yard. 

37.  Reduce  3ft.  4'  8"  3'"  to  feet  and  decimals  of  a  foot. 

CASE   II. 

165.    To  find  the  value  of  a  decimal  in  integers  of  less  denomi- 
nations. 

1.  What  is  the  value  of  .832296  of  a  £  ? 

Analysis. — Fu-st  multiply  the  decimal  operation. 

by  20,  which  brings  it  to  the  denomination  .832296 

of  shillings,  and  after  cutting  off  from  the  20 

right  as  many  places  lor  decimals  as  there  16.645920 

are  in  the  given  nunlber,  we  have  1 65.  and  1 2 

the  decimal   .645920   oter.     This   is  re-  7.751040 

duced  to  pence  by  multiplying  by  12,  and  4 

then  to  farthings  by  multiplying  by  4.  3.004160 

Ans.  165.  "id.  3far. 

165.  How  do  you  find  the  value  of  a  decimal  in  integers  of  less  denomi 
nations  1 


174r  DKNOMINATE    DECIMALS. 

Rule. — I.  Mulliply  (he  decimal  by  the  uiiis  of  the  scaCe 
which  connect  it  with  the  next  less  denomination,  pointing  off  as 
in  the  multijjli cation  of  decimals. 

II.  Mullijily  the  decimal  part  of  the  product  as  before,  and 
tontinue  so  to  do  until  the  decimal  is  reduced  to  the  required  deno- 
minations.     The  integers  cut  off  at  the  left  form  the  ansuer. 

EXAMPLES. 

1.  "What  is  the  value  of  .6725  of  a  hundred  weight  ? 

2.  What  is  the  value  of  .61  of  a  pipe  of  wine  ? 

3.  What  is  the  value  of  .83229  of  a  £  ? 

4.  Required  the  value  of  .0625  of  a  barrel  of  beer. 

5.  Requii'ed  the  value  of  .42857  of  a  month. 

6.  Required  the  value  of  .05  of  an  acre. 

7.  Required  the  value  of  .3375  of  a  ton. 

8.  Required  the  value  of  .875  of  a  pipe  of  wine. 

9.  What  is  the  value  of  .375  of  a  hogshead  of  beer  ? 

10.  What  is  the  value  of  .911111  of  a  pound  troy? 

11.  What  is  the  value  of  .675  of  an  English  ell  ? 

12.  What  is  the  value  of  .001136  of  a  mile  in  length  ? 

13.  What  is  tire  value  of  .000242  of  a  square  mile  ? 

14.  Required  the  value  of  .4629  degrees. 

15.  Required  the  value  of  .875  of  a  yard. 

16.  Required  the  value  of  .3489  of  a  pound  apothecaries. 

17.  Required  the  value  of  .759  of  an  acre. 

18.  Required  the  value  of  .01875  of  a  ream  of  paper. 

19.  Requii'ed  the  value  of  .0055  of  a  ton. 

20.  Required  the  value  of  .625  of  a  sliilling. 

21.  Required  the  value  of  .3375  of  an  acre. 

22.  Required  the  value  of  .785  of  a  year,  of  3651  daya 


REPEATING    DECIMALS.  175 


CIRCULATING  OR  REPEATING  DECIMALS. 

166.  In  changing  a  common  to  a  decimal  fraction,  there  are 
two  general  cases  : 

1st.  When  the  division  terminates  ;  and 
2d.  When  it  does  not  terminate. 

In  the  first  case,  the  quotient  will  contain  a  limited  number 
of  decimal  places,  and  the  value  of  the  common  fraction  will  be 
exactly  expressed  decimally. 

In  the  second  case,  the  quotient  will  contain  an  infinite  num- 
ber of  decimal  places,  and  the  value  of  the  common  fraction  can- 
not be  exactly  expressed  decimally. 

CASE   I. 

167.  When  the  division  terminates  : 

When  a  common  fraction  is  reduced  to  its  lowest  terms  (which 
we  suppose  to  be  done  in  all  the  cases  that  follow),  there  wuU 
be  no  factor  common  to  its  numerator  and  denominator  (Art. 
120). 

1.  Reduce  ^  to  its  equivalent  decimal. 

Analysis. — Annexing  one  decimal  0  to  the  operation. 

numerator  multiplies  it  by  10,  or  by  2  and  5;  50)17. 00(. 34 

hence,  2  and  5  become  prime  factors  of  the  nu-  15  0 

merator  every  time  that  an  0  is  annexed.     But  2  00 

if  the  division  be  exact,  these  prime  factors,  and  2  GO 
none  others,  must  also  he  found  iii  the  denomi- 
nator (Art.  91). 


166.  How  many  cases  are  there  in  chanfring  a  vulgar  to  a  decimal  frac- 
tion 1     What   are  they  ?  .  What  distinguishes  one  of  these  cases  from  tho 

other  1 

167.  How  do  you  determine  when  a  vulgar  fraction  can  be  exactly  ex- 
pressed decimally  ^  How  many  decimal  places  will  there  be  in  the 
ciuotient  ' 


176  CIRCULATING    OR 

OPERATION. 

2.  Reduce  -^  to  its  equivalent  decimal.        35)  50  (.1388  + 

Analysis. — 36  =  18x2  =  9X2x2=  ?^ 

3  X  3  X  2  X  2  :  in  which  we  see  that  the  de-  ^'^^ 

nominator  contains  other  factors  than  2  and  o.  ^^^ 
aud  liencc,  the  fraction    cannot   be   exactly  ex-  "^-^ 

pressed  bij  decimals  (An.  91).      Hence,  to  deter-  ^^^ 

mine  whether  a  common  fraction  can  be  exactly  ^^^ 

expressed  decimally :  288 

I.  Decompose  the  denominator  into  its  prime  factors;  and  if 
there  are  nn  factors  other  than  2  and  5,  (he  exact  division 
can   be  made  : 

II.  If  there  are  other  prime  factors,  the  exact  division  cannot 

he  made. 

Note.— Every  decimal  0  annexed  to  the  numerator,  introduces  the 
two  factors  2  and  5  ;  and  these  factors  must  be  introduced  until  we 
have  as  many  of  each  as  there  are  in  the  denominator  after  it  shall 
have  been  decomposed  into  its  prime  factors  2  and  5.  But  the  quo- 
tient will  contain  as  many  decimal  places  as  there  arc  decimal  O's  in 
the  dividend  (Art.  155);  hence, 

The  number  of  decimal  jylaces  in  the  quotient  will  be  equal  to 
ike  greatest  number  of  equal  factors  2  or  5,  in  the  divisor. 

EXAJrPLES. 

1.  Can  ^^3- be  exactly  expressed  decimally  ?       opkration. 
how  many  jdaces  ?  25)  70  (.28 

50 

25  =  5  X  5  ;    hence,   the  fraction  can  be  ex-  200 

actly  expressed  decimally.  200 

Find  the  decimals  and  number  of  places  in  the  following: 

1.  Express  ^?^  decimally.  5.  Express  ^Vo  decimally. 

2.  Express  ^-/^  decimally.  G.  Express  -^^  decimally. 

3.  Express  ^y^y  decimally.  7.  Express  ^f^  decimally. 

4.  Express  yi^^  decimally.  J     8.  Express  /^j  decimally. 


REPEATING   DECIMALS.  177 

CASE   II. 
168.    When  (he  division  does  not  terminate. 

1.  Let  it  be  required  to  reduce  i  to  its  equivalent  decimal. 

Analysis. — By  annexing  decimal  ciphers  to 
the  numerator  1,  and  making  the  division,  wc  operation. 

find  the  equivalent  decimal  to  be  .3333  +,  &c.,  3)1  0000 

giving   3's  as  far  as  avc  choose  to  continue  the  .3333  + 

divifcion. 

The  further  the  division  is  continued,  the  nearer  the  value  of 
the  decimal  will  approach  to  ^,  the  true  vaZ;/eof  the  common 
fraction.  We  express  this  approach  to  equality  of  value,  by 
saying,  that  if  the  division  be  continued  without  limit,  that  is,  to 
infill  ity,  the  value  of  the  docimal  will  then  become  equal  to  that 
of  the  common  fraction  ;  thu  s, 


.33.33  +,  continued  to  infinity 


1 . 

3  ' 


for,  every  annexation  of  a  3  brings  the  value  nearer  to  ^. 

Also,  .9999  +,  continued  to  infinity  =  1 ; 

for,  every  annexation  of  a  9  brings  the  value  nearer  to  1. 

2.  Find  the  decimal  corresponding  to  the  common  fraction  |^. 

Analysis. — Annexing  decimal   ciphers    and         operation. 
dividing,  we  find  the  decimal  to  be  .2222  +,  in  9)2.0000 


which  we  see  that   the  figure  2  is  continually  .2222  + 

repeated. 

169.  A  Circulating  Decijial  is  a  decimal  fraction  in 
which  a  single  figure,  or  a  set  of  fiigures,  is  constantly  repeated. 

170.  A  Repetend  is  a  single  figure  or  a  set  of  figures,  which 
is  constantly  repeated. 

168.  Can  one-third  be  exactly  expressed  decimally'!  What  is  the  foni) 
of  the  quotient  1  To  what  docs  the  value  of  this  quotient  approach  ? 
When  does  it  become  equal  to  one-third  1 

169.  What  is  a  circulatinjr  decimall 

170.  What  is  a  repetcnd  1 


178  CIKCr LATINO    OK 

171.  A  Single  Repetend  is  one  in  -which  only  a  single 
figure  is  repeated  ;  as 

I  =  .2222+,     or     |  =  .3333 +. 

Such  repetends  are  expressed  by  simply  putting  a  mark  over 
the  first  figure  ;  thus, 

.2222  +,  is  denoted  by  .%  and  .3333  +  by  .'3. 

172.  A  Compound  Repetend  has  the  same  set  of  figures 
circulating  alternately ;  thus. 

If  =  .57  57  +,  and  -|4||  =  .5723  5723  +, 
are  compound  repetends,  and  are  distinguished  by  marking  the 
first  and  last  figures  of  the  circulating  period.     Thus,  .57  57  + 
is  written  .'57',  and  .5723  5723  +  is  written  .'5723'. 

173.  A  Pure  Repetend  is  one  which  begins  with  the  first 
decimal  figure ;  as 

.'3,         .'5,         .'473',         &c. 

174.  A  Mixed  Repetend  is  one  which  has  significant  figures 
or  ciphers  between  the  repetend  and  the  decimal  point ;  or 
which  has  whole  numbers  at  the  left  hand  of  the  decimal  point ; 
such  figures  are  caWo.^  finite  figures.     Thus, 

.0733',     .473',     .3'573',     G.'o, 

are  all  mixed  repetends  ;   .0,  .4,  .3,  and  6,  are  ih^  finite  figures. 

175.  Similar  Repetends  are  such  as  begin  at  equal  dis- 
tances from  the  decimal  points  ;  as  .3'54',  2.7'534;.'. 

176.  Dissimilar  Repetends  are  such  as  begin  at  different 
^.'stances  from  the  decimal  point ;  as  .'253',  .47'52'. 

177»  Conterminous  Repetends  are  such  as  end  af  equal 
distances  from  the  decimal  points  ;  as  .1'25',  .'354'. 


171.  ^Vhat  is  a  single  repetend  ? 

372.  What  is  a  compound  repetend  T 

173.  What  is  a  pure  repetend  1 

174.  What  is  a  mixed  rcj)etcnd  1 

175.  What  are  similar  repetends  t 
17().  What  are  dissimilar  repctoudal 


REPEATING  .DECIMALS.  179 

178.  Similar  and  Conterminous  Repetends  are  sucli  as 
bei^in  and  end  at  the  same  distances  from  the  decimal  point ; 
thus,  53.2753',  4.6'325',  and  .4^632',  are  similar  and  conter- 
minous repetends. 

REDUCTION  OF  REPETENDS  TO  COMMON  FRACTIONS. 

CASE   I. 

179.  To  reduce  apure  repetendto  its  equivalent  common  fraction. 

Analysis, — This  proposition  is  to  be  analyzed  by  examining  the 
law  of  forming  the  repetends. 

1st.  .|  =.llll+&c.  =  .M;  and    f  =.4444  +  &c.  =  .M  : 

2d.    ^=.010101+&:c  =>01';       and  f|- =.2727+&c.  =  .'27' : 
3d.  ^1^=^.001001 +Scc.  =  .^001';  and  fff  =  .324324  +  &c.  =  .'324'. 
&c.  &:c.  &:c.  &c. 

The  above  law  for  the  formation  of  repetends  does  not  depend  on 
the  multipliers  4,  27,  and  324,  but  would  be  the  same  for  any  other 
figures;  hence, 

The  value  of  any  jmre  repetend  is  equal  to  the  7iiimher  denot- 
ing  the  repetend,  divided  hy  as  many  9's  as  there  are  figures. 

EXAMPLES. 

1.  What  is  the  equivalent  common  fraction  of  the  repetend 
0.3?^ 

Now,  9  =  i  =  0.33333  +.=  0.^3. 

2.  What  is  the  equivalent  common  fraction  of  the  repetend 
:i62'  ? 

We  have,  gf  I  =  t'tt  -^"*' 

3.  What  are  the  simplest  equivalent  common  tractions  of  the 
repetends  .^6,.^162',   0.769230',  .^945',  and  .^09'? 

4.  AVhat  are  theleastequivalentcommonfractionsof  the  repe- 
tends .^594405',  .^36',  and  .^142857'  ? 


177.  What  are  conterminous  repetends  \ 

178.  What  are  similar  and  conterminous  repetends  1 

17'J.  How  do  you  find  a  common  fraction  equivalent  to  a  puie  repetend  ! 


180  CIRCULATING    OR 

CASE    II. 

180.  To  reduce  a  mixed  repelend  to  its  cquivalen  t  common  fraction. 

Analysis. — A  mixed  repetend  is  composed  of  the  finite  figures 
wliich  precede,  and  of  tlie  repetend  itself;  hence,  its  value  must  be 
equal  to  such  finite  figures  plus  the  repetend. 

When  the  repetend  begins  at  the  decimal  point,  the  unit  of  the  first 
figure  is  .1.  But  if  the  repetend  begins  at  any  place  at  the  right  of 
the  decimal  point,  the  unit  value  of  the  first  figure  will  be  diminished 
ten  limes  for  each  place  at  the  right,  and  hence,  O's  must  be  annexed 
to  the  9"s  Avhich  form  the  divisor  ;  therefore, 

To  the  finite  figures,  add  the  repetend  divided  hy  as  many  9'* 
as  it  contains  places  of  figures,  ivith  as  many  O's  annexed  to 
them  as  there  are  places  of  decimal  figures  preceding  the  repe- 
tend ;  the  sum  reduced  to  its  simplest  form  icill  be  the  equivalent 
fraction  sought. 

EXAMPLES. 

1.  Required  theleast  equivalent  common fractioQ  of  the  mixed 
repetend,  2.4:'18'. 

Now, 

2.4^18'  =  2  +  T-V  +^18'  =  2  +  T^  +  J4  =  2f 3.  Ans. 

2.  Required  theleast  equivalent  common  fraction  of  the  mixed 
repetend  .o'925'. 

We  have,         .5^925'  =  -^  +  ^^  =  If-  ^««- 

3.  What  is  the  least  equivalent  common  fraction  of  the  repe- 
tend .008^497133'? 

Wo  havf        008M971?!i'  a        i         497133     S3_, 

nCIKHt,        .UUOiJ/IOO     —  Yoo^   i-    99-9999-000-   —  TTTaS' 

4.  Required  the  least  equivalent  common  fractions  of  the  mixed 
repetends  .13^8,  7.5^43',  .04'3o4',  37.5^4,  .G7o',  and  .7^:i4347'. 

5.  Required  the  least  equivalent  common  fractions  of  tlie  mixed 
rcpetends  0.7^5,  0.4^38',  .09^3,  4.7^543',  .009'87^  and  .4'o. 


180.  How  do  you  find  the  value  of  a  mixed  repetend  ' 


REPEATING    DECIMALS..  181 

CASE    III. 

181.    To  find  the  finite  figures  and  the  repetend  corresponding 
to  Q,ny  common  fraction. 

1.  Find  the  finite  figures  and  the  repetend  corresponding  to 

the  fraction  ^^^q- 

Analysis. — 1st.  Reduce  the  frac- 
tion to  its  lowest  terms,  and  then 


OPKRATION. 
fi  3 


560       280 
3 


280        2X2X2X5X7 
280)3.000  +(.010^714285' 


find   all   the  factors  2  and  5  of  the 
denominator. 

2d.  Add  decimal  ciphers  to  the 
numerator  and  make  the  division. 

3d.  The  number  of  finite  decimals  preceding  the  first  figure  of  tho 
repetend  will  be  equal  to  the  greatest  number  of  factors  2  or  5 
(Art.  167).     In  this  example  it  is  3. 

4th.  When  a  remainder  is  found  which  is  the  same  as  a  previous 
dividend,  the  second  repetend  begins. 

5th.  The  number  of  figures  in  any  repetend  will  never  exceed  tha 
number,  less  l ,  of  the  units  in  that  factor  of  tlie  denominator  which 
does  not  contain  2  or  5.  Tn  the  example,  that  number  is  7,  and  the 
number  of  figures  of  the  repetend,  is  6.     Hence, 

Divide  the  numerator  of  the  common  fraction,  reduced  to  its 
lowest  terms,  hj  the  denominator,  and  point  off  in  the  quotient 
the  finite  decimals,  if  any,  and  the  repetend. 

EXAMPLES. 

1.  Required  to  find  whether  the  decimal,  equivalent  to  the 
common  fraction  3-f  ^-qt?  is   finite   or  c 
finite  figures,  if  any,  and  the  repetend 

Analysis. — We  first  reduce  the 
fraction  to  its  lowest  terms,  giving 
TtI It-  ^^°  ihen  search  for  the  fac- 
tors 2  and  5  in  the  denominator, 
and  find  that  2  is  a  factor  3  times  ; 

hence,  we  know  that  there  are  three      Q'^f;«")»'3  ofl  +  C  on8''4Q7l  3*?' 
finite  decimals  preceding  the  repe- 
tend.   We  next  divide  the  numerator  83  by  the  denominator  9768,  and 

181.  How  do  you  find  the  finite  figures  and  the  repetend  corres]>ondi!ii» 
to  any  CDUuniMi  fr;icli()ii  ? 


jirculating :    required 

the 

OPERATION. 

249 

83 

29304 

9768 

83 

83 

9768 

2X2X2X1221 

182  PKOPERTIES    OF    THE 

note  that  the  repeteiid  begins  at  the  fourth  place.  After  the  ninth 
'livision,  .we  find  the  remainder  83  ;  at  this  point  the  figures  begin  to 
'epeat ;  hence,  the  rcpetend  has  6  places. 

2.  Find  the  finite  decimals,  if  any,  and  the  rcpetend,  if  any, 
)f  the  fraction  fy^* 

3.  Find  the  finite  decimals,  if  any,  and  the  repetend,  if  any, 
tf  the  fraction  ■.  A„-. 

4.  Find  the  finite  decimals,  if  any,  and  the  repetend,  if  any, 
of  the  fractions  ^22^   _^^  _7_2_. 


PROPERTIES  OF  THE  REPETENDS. 

182.  There  ai-e  some  properties  of  the  vepetends  which  it  is 
important  to  remark. 

1.  Any  finite  decimal  may  be  considered  as  a  circulating 
decimal  by  making  ciphers  to  recur  ;  thus, 

.35  =  .35^0  =  .35^00'  =  .35^000'  :=  .35^0000',  &c. 

2.  If  any  circulating  decimal  have  a  repetend  of  any  number 
of  figures,  it  may  be  changed  to  one  having  twice  or  thrice  that 
number  of  figures,  or  any  multiple  of  that  number. 

Thus,  a  repetend  2.3^57'  having  two  figures,  may  be  changed 
to  one  having  4,  6,  8,  or  10  places  of  figures.     For, 

2.3^57'  =  2.3^5757'  =  2.3^575757'  =  2.3^57575757',  &c.; 

so,  the  repetend  4.16^31G'  may  be  written 

4.16^316'  =  4.1G^31G316'  =  4.16^310316316',  Sec.  &c.; 

and  the  same  may  be  shown  of  any  other.  Hence,  two  or  more 
repetends,  having  a  different  number  of  places  in  each,  may  be 
reduced  to  repetends  having  the  same  number  of  places,  in  the 
following  manner : 

182.  How  may  a  finite  decimal  be  made  circulating  1  When  a  repetend 
has  a  given  number  of  places,  to  what  otlicr  farm  may  it  be  reduced! 
How  !  Into  wiiat  form  may  any  circulating  dociniai  be  transformed  ?  To 
wh.%1  fvnu  may  two  or  more  icjielcnds  be  reduced  1 


KEPETENDS.  183 

Find  the  least  comvion  multiple  of  the  number  of  places  in 
each  repetend,  and  reduce  each  repetendto  such  number  of  places. 

Ex.  1.  Reduce  .13^8,  7.5^43^  .04^354',  to  repetends  having 
the  same  number  of  places. 

Since  the  number  of  places  are  now  1,  2,  and  3,  the  least 
common  multiple  is  6,  and  hence  each  new  repetend  Avill  con- 
tain 6  places  ;  that  is, 

.13^8=. 13^888888';  7.5^43'.zr 7.5^434343';  and 
0.4^354'  =  0.4^3o43o4'. 

Ex.  2.  Reduce  2.4^8',  .5^925',  .008^497133',  to  repetends 
having  the  same  number  of  places. 

3.  Any  circulating  decimal  may  be  transformed  into  another 
having  finite  decimals  and  a  repetend  of  the  name  number  of 
figures  as  the  first.     Thus, 

.^57'  =  .575'  =:  .57\57'  =  .57575'  =  .5757^57';  and 
3.4785'  =  3.47^857'  =  3.478^578'  ==  3.4785785'; 
and  hence,  any  two  repetends  may  be  made  similar. 

These  properties  may  be  proved  by  changing  the  repetends 
mto  their  equivalent  common  fractions. 

4.  Having  made  two  or  more  repetends  similar  by  the  last 
article,  they  may  be  rendered  conterminous  by  the  pi-evious 
one ;  thus,  two  or  more  repetends  may  alicays  be  made  similar 
and  contenninous. 

1.  Reduce  the  circulating  decimals  165.'164',  .'04',  .037  to 
such  as  are  similar  and  conterminous. 

2.  Reduce  the  circulating  decimals  .5'3,  .475',  and  1.'757', 
to  such  as  are  similar  and  conterminous. 

5.  If  two  or  more  circulating  decimals,  having  several  repe- 
tends of  equal  places,  be  added  together,  their  sum  will  have  a 
repetend  of  the  same  number  of  places ;  for,  every  two  sets  of 
repetends  toill  give  the  same  sum. 

6.  If  any  circulating  decimal  be  multiplied  by  any  number, 
the  product  will  be  a  circulating  decimal  having  the  same 
number  of  places  in  the  repetend ;  for,  each  repetend  will  be 
taken  the  same  number  of  times,  and  consequently  must  j^i'oduce 
the  same  product. 


184  ADDITION     OF 

ADDITION  OF  CIRCULATING  DECIMALS. 

183.  To  add  circulating  decimals  : 

I.  Make  the  reijetends,  in  each  number  to  be  added,  similar 
and  conterminous. 

II.  Write  the  places  of  the  same  unit  value  in  the  same  column, 
and  60  mani/  figures  of  the  second  repelend  in  each  as  shall 
indicate  with  certainty,  how  many  are  to  he  carried  from  one 
repdend  to  the  other :  then  add  as  in  whole  numbers. 

Note — If  all  the  figures  of  a  repetend  are  9's,  omit  them  and  add 
to  the  figure  next  at  the  left. 

EXAMPLES. 

1.  Add  .1*2^5,  4.^163',  r.7143',  and  2.^54'  together. 
Dissimilar.     Similar.  Similar  and  Conterminous. 

.12'5  —    .12'5        =    .12'555oo5555555'  -  -  -  5555 

4.'163' =4.16^316'   =  4.16^310316316310'  -  -  -  3103 

1.7143'  =  1.71^4371'=  1.71^437143714371'  -  -  -  4371 

2.'54'  =  2.54^54'     =  2.54^545454545454'  -  -  -  5454 

The  true  sum  z=  8.54^854470131097'  1  to  carry. 

2.  Add  G7.3'45',  9.^051',  .^25',  17.47,  :5,  together. 

3.  Add  ;475',  3.75^43',  04.75',  .^57',  .1788',  together. 

4.  Add  :o,  4.37,  49.4\57',  .4^954',  .7345',  together. 

5.  Add  .^175',  42.^57',  .3753',  .4'954',  37^54',  together. 

6.  Add  105,  .\1W,  147.^04',  4.^95',  94.37,  4.712345'  to- 
gether. 

SUBTRACTION  OF  CIRCULATING  DECIMALS. 
184.  To  subtract  one  circulating  decimal  from  another. 
I.  3fuke  the  repetends  similar  and  co7iterminous. 
'  II.   Subtract  as   in  finite   decimals,  observing  that   when    the 
repetend  of  the  lower  line  is  the  larger,  1  must  be  carried  to  the 
first  rigid  hand  figure. 


183.  How  do  you  add  Circulating  Decimals  ? 
IS'l.  How  do  you  subtract  Circulating  Decimals  ! 


CIRCULATING    DECIMALS. 


186 


EXAMPLES. 

1.  From  11.475'  take  3.45735'. 

Dissimilar.       Similar.     Similar  and  Conterminous. 

11.^75'  =:  11.47^57'  ==  11.47^575757'    -     -     -     ■ 
3.45735'  =  3.45735'  =    3.45735735'    -     -     -     . 

The  true  difference  =    8.01^840021'  1  to  carry. 


575 

735 


2.  From  47.5^3  take  1.757'. 

3.  From  17.^573'  take  14.57. 

4.  From  17.4^3  take  12.34^3. 

5.  From  1.12754' take. 47384'. 


G.  From  4.75  take  .37^5. 

7.  From  4794  take  .1744'. 

8.  From  1.457  take  .3654. 

9.  From  1.4^937' take  .1475. 


MULTIPLICATION  OF  CIRCULATING  DECIMALS. 

185.  To  multiply  one  circulating  decimal  by  another. 

Change  the  circulating  decimals  into  their  equivalent  common 
fractions.,  and  then  multiply  them  together ;  then,  reduce  the 
product  to  its  equivalent  circulating  decimal. 

EXAMPLES. 

1.  Multiply  4.25^3  by  .257. 


OPERATION. 


4  95^3    —   4-l-_25_4.  _3_   —    4  4.    225      I 


3 

900 


2  2_R    3  S28 

900    —     9T0~ 


45  0 


9.A1 
225* 


Also,  .257  =:  -rsoo  5  lience, 

-   957  y  _2_5J_  _  2  45  9A9  _  1  0931 OT)  • 
225  '^  1000  —  225000  —  x.v^UAVU, 

and  since  225000  =  5x5x5x5x5x2x2x2x9; 
there  will  be  five  places  of  finite  decimals,  and  one  figure  in 
the  repetend  (Art.  1G7). 

NoTK. — Much  labor  will  be  saved  in  this  and  the  next  rule  by  Icccpinp^ 
every  fraction  in  its  lowest  terms  ;  and  when  two  fractions  are  to  be 
vndtipHcd    together,  cancel  all  the  factors  common   to  both  terms  be 
fore  niakiyig  the  multiplicalion. 


]Sr>.   How  do  you  mulliply  Circulating  Decimals  1 

9 


186 


i)ivisio>r  OF 


2.  Multiply  .375'4  by  14.75. 

3.  Multiply  .4'253'  by  2.57. 

4.  Multiply  .437  by  3.7^5. 

5.  Multiply  4.573  by  .375'. 


6.  Multiply  3.45^ 6  by  .42^5. 

7.  Multiply  1.^456'  by  4.2^3. 

8.  Multiply  45.1^3  by  .^245'. 

9.  Multiply .4705^3  by;.7^35'. 


DIVISION  OF  CIRCULATING  DECIMALS. 

186.  To  divide  one  circulating  decimal  by  another. 

Change  the  decimals  into  their  equivalent  common  fractions, 
and  find  the  quotient  of  these  fractions.  Then  change  the  quch 
tieni  into  its  equivalent  decimal. 


EXAMPLES. 


1.  Divide  5G.^6  by  137. 


OPERATION. 


56.^6 


Then, 


]  7  0 
3 


56  4-  6-  —  SJLO  _  no.. 
137  =  ip  X  t1_  =  i7fi  ^  /413G2530'. 


4]  1 


2.  Divide  24.3^18'  by  1.792. 

3.  Divide  8.59G8  by  .2^45'. 

4.  Divide  2.295  by  .^297'. 

5.  Divide  47.345  by  1.76'. 


0.  Divide  13.5UG9533  by  4.^297' 

7.  Divide  .H5' by  .^118881'. 

8.  Divide  .^475'  by  .3753'. 

9.  Divide  3.753'  by  .^24'. 


CONTINUED    FRACTIONS. 

1.  If  -we  take  any  irreducible  fraction,  as  J-|,  and  divide  both 
terms  by  the  numerator,  it  will  take  the  form 

15       1        1  ,  ,.        ,      ^.   .  . 

ml  =  V9  =  T    I    ]4>  ^y  malcmg  the  division. 

ZJ  1..  ^   "I      15 

If  now,  we  divide  both  terms  of  ^|^  by  the  numerator  14,  we 

have 

14       1 


15  ~  1  +  tL 


]8G.  lluw  <lo  you  divide  Circulating  Deciiu.ils  ? 


CTKCULATING    DECi:V[ALS.  187 

1 


If,  now,  we  replace  \i  by  its  value, —,  we  s 

15         1 


shall  have 


2d       1  +  1 


1  -I-  -J~  • 

a  fraction  of  this  form  is  called  a  coniimied  fraction  ;  hence, 

187.  A  Continued  Fraction  has  1  for  Us  numerutor,  and 
for  its  deno)ninator  a  ivhole  manber  plus  a  fraction  which  also 
has  a  numerator  of  1,  and  for  a  denominator,  a  whole  number 
plus  a  similar  fraction,  and  so  on. 

2.  Reduce  \^  to  the  form  of  a  continued  fraction. 


hence, 


15 

1 

4        1 

3       1 

19 

"l+T^/ 

15-3  +  1 

'    4~l+i 

15 
19 

1 

1  +  1 

3  +  1 

1 

+  i- 

luce 

!  ^^  to  the  form  of  a  continued  fraction. 

829 
347  ~ 

2  +  1 

2  +  1 

1  + 

1 

1  +  1 

3  +  tV 
4.  Reduce  -^-Ar  to  the  form  of  a  continued  fraction. 

^_  1 

149-2  +  1 


3  +  1 


2+  1 


2  +  1 


1  +  h 


'B7.  What  is  a  Continued  Fraction  1 


188  CONTINUED    FKACTIONS. 

Note. — In  a  similar  manner,  any  irreducible  common  fraction  may 
be  placed  under  the  i'orm  of  a  continued  fraction. 

188.  Let  us  now  consider  the  last  example.     The  fractions, 
11  1 


2'     2  +  1'     2  +  1' 


&c., 


3  +  1, 
are  called,  the  Jirst,  second,  third,  &c.,  approximating  fractions; 
and  the  object  in  placing  a  common  fraction  under  the  form  of  a 
continued  fraction  is  to  find  its  api^roximate  value. 

If  we  stop  at  the  first  approximating  fraction,  1,  the  denomi- 
nator 2  will  be  less  than  the  true  denominator ;  hence,  the  value 
of  the  first  ajjproximating  fraction  will  be  too  great ;  that  is,  it 
will  exceed  the  value  of  the  given  fraction. 

If  we  stop  at  the  second,  the  denominator  3  will  be  less  than 
the  true  denominator  ;  hence,  1  will  be  greater  than  the  number 
to  be  added  to  2 ;  therefore,  2  +  l  is  too  small,  and  1  -r-  1  +  ^ 
is  too  large :  that  is,  it  is  greater  than  the  value  of  the  given 
fraction. 

Thus,  every  odd  approximating  fraction  gives  a  value  too 
large,  and  every  even  one  gives  a  value  too  small. 

EXAMPLES. 

1.  Place  §1^  under  the  form  of  a  eontinued  fraction,  and  find 
the  value  of  each  of  the  approximating  fractions. 

2.  Place  1^  under  the  form  of  a  continued  fraction,  and  find 
the  value  of  each  of  the  approximating  fractions. 

3.  Place  11  under  the  form  of  a  continued  fraction,  and  find 
the  value  of  each  approximating  fraction. 

4.  Place  -I-  under  the  form  of  a  continued  fraction,  and  find 
the  value  of  each  approximating  fraction. 

5  Place  -^-1  under  the  form  of  a  continued  fraction,  and  find 
.he  value  of  each  approximating  fraction. 

1S3.  ^^'hat  is  an  nj)[)r<i\imating  fraction!  Is  the  first  aiiproximaling 
Taction  too  large  or  too  small!  How  is  the  second  !  How  are  all  the 
xld  ones  1     How  are  all  the  cvoii  ones  1 


RATIO    AXD    PROPORTION'.  189 

RATIO    AND    PROPORTION. 

189.  Proportion  is  the  relation  whicli  one  number,  re- 
garded as  a  standard,  bears  to  another,  with  respect  either  to 
their  difference  or  quotient.  Two  numbers  may  be  compared  in 
two  ways  : 

1st.  By  considering  hoio  much  one  is  greater  or  less  than 
the  other,  which  is  shown  by  their  difference  ;  and, 

2d.  By  considering  Jiow  many  times  one  is  contained  in  the 
other,  which  is  shown  by  their  quotient. 

In  comparing  two  numbers,  one  with  the  other,  by  means  of 
their  difference,  the  less  is  always  taken  from  the  greater. 

In  comparing  two  numbers,  one  with  the  other,  by  means  of 
their  quotient,  one  of  them  must  be  regarded  as  a  standard 
which  measures  the  other,  and  the  quotient  which  arises  by 
dividing  by  the  standard,  is  called  the  ratio. 

190.  Every  ratio  is  derived  from  two  terms  :  the  first,  or  the 
standard  is  called  the  antecedent,  because  it  is  supposed  to  be 
known  beforehand  ;  and  the  second,  the  consequent ;  and  the 
two,  taken  together,  are  called  a  couplet.  The  antecedent  will 
be  regarded  as  the  standard. 

If  the  numbers  3  and  12  be  compared  by  their  difference,  the 
result  of  the  comparison  will  be  9  ;  for,  12  exceeds  3  by  9.  If 
they  are  compared  by  means  of  their  quotient,  the  result  will 
be  4 ;  for,  3  is  contained  in  12,  4  times ;  that  is,  3  measuring 
12,  gives  4. 

191.  The  ratio  of  one  number  to  another  is  expressed  in  two 
■ways  : 

1st.  By  a  colon;  thus,  3  :  12 ;  and  is  read,  3  is  to  12;  or, 
3  measurins;  12. 

189.  In  how  many  ways  maj'  wo  numbers,  having  the  same  unit,  be 
compared  with  each  other  1  If  you  compare  by  their  difference,  what 
do  you  da  !  If  you  compare  by  the  quotient,  how  do  you  regard  one  of 
the  numbers  ''     What  is  the  ratio  1 

190.  From  how  many  terms  is  a  ratio  derived  !  What  is  the  first  term 
called  1     What  is  the  second  called  I     Which  is  the  standard  1 

191.  How  may  the  ratio  of  two  number-s  be  expre.ssed  1     How  read  1 


190 


RATIO    AND     PROPORTION. 


2d.  In  a  fractional  form,  as  Jj^-;  or,  3  measuring  12. 

192.  If  two  couplets  have  tbe  same  ratio,  their  terms  ara 
said  to  be  proportional :  the  couplets, 

4     :     20     and     1     :     5, 

have  the  same  ratio  5 ;  hence,  the  terras  are  proportional,  and 

are  written, 

4     :     20     :  :     1     :     5, 

by  simply  placing  a  double  colon  between  the  couplets.     The 
terms  are  read, 

4  is  to  20     as     1  is  to  5, 

and  taken  together,  they  are  called  a  projiortimi :  hence, 

A  projwrtion  is  an  expression  of  equality  between  two  ratios. 

What  are  the  ratios  of  the  proportions  ? 


6 

24         : 

8         : 

32 

9 

36         : 

10         : 

40 

8 

72         : 

12         : 

108 

4 

48         : 

5         : 

60 

193.  The  1st  and  4th  terms  of  a  proportion  are  called  the  ex- 
tremes ;  the  2d  and  3d  terms,  the  means.    Thus,  in  any  proportion, 

6     :     24     :  :     8     :     32, 
6  and  32  are  the  extremes,  and  24  and  8  the  means  : 

24       32 
Smce  -=-, 

we  shall  have,  by  reducing  to  a  common  denominator, 

24  X  8  _  32  X  6 
6  X  8""    6x8' 
But  since  the  fractions  are  equal,  and  have  the  same  denorai- 
nators,  their  numerators  must  be  equal,  viz. : 

24  X  8  =  32  X  6  ;  that  is, 


192.  If  two  couplets  have  the  same  Peitio,  what  is  said  of  the  terms  ! 
How  arc  thoy  written  T     How  read  ^     What  is  a  proportion  ? 

193.  Which  are  the  extremes  of  a  proportion  \      Which   the  means  ' 
What  is  the  product  of  the  extremes  equal  to  1 


KATlU    AND    PKOPORTIOJ^.  191 

In  any  projwrtion,  ike  product  of  Ike  extremes  is  equal  to  the 
product  of  the  means. 

Thus,  in  the  proportions, 

1     :       8     :  :     2     :     16;  we  have  1  x  16  =    2x8. 
4     :      12     :  :     8     :     24;     "       "       4  X  24  =:  12  X  8 ; 

194.   Since,  in  any  proportion,  the  product  of  the  extremes  is 
equal  to  the  product  of  the  means,  it  follows  that, 

1st.  //'  the  product   of  lite   means   be  divided   by  one  of  the 
extremes,  the  quotient  will  be  the  other  extreme. 

Thus,  in  the  proportion, 

4     :     16     :  :     6     :     24,  and  4  x  24  =  16  X  6  =  96  ; 
then,  if  96,  the  product  of  the  means,  be  divided  by  one  of  the 
extremes,  4,  the  quotient  will  be  the  other  extreme,  24 ;  or,  if 
the  product  be  divided  by  24,  the  quotient  will  be  4. 

2d.  //  the  product  of  the  extremes  be  divided  by  either  of  the 
means,  the  quotient  will  be  the  other  mean. 

Thus,  if4x24=:16x6=:96be  divided  by  16,  the  quo- 
tient will  be  6  ;  or  if  it  be  divided  by  6,  the  quotient  will  be  16. 

EXAMPLES. 

1.  The  first  three  terms  of  a  proportion  are  5,  10,  and  19  ? 
what  is  the  fourth  terra  ? 

2.  The  first  three  terras  of  a  proportion  are  6,  24,  and  14 : 
what  is  the  fourth  terra  ? 

3.  The  first,  second  and  fourth  terms   of  a  proportion  are 
9,  12  and  16  :  what  is  the  third  term  ? 

4.  The  first,  third   and  fourth  terms  of  a  proportion  are  16, 
8,  and  20  :  what  is  the  second  term  ? 

5.  The  second,  third    and  fourth  terms  of  a  proportion  are 
48,  90,  and  45  :  what  is  the  first  term  ? 


194.  If  the  product  of  the  means  be  divided  by  one  of  tlie  extremes, 
whnt  will  the  quotient  be  ]  If  the  product  of  the  means  be  divided  by 
either  extreme,  what  will  the  quotient  be  ? 


192  KATIO    AX3)    PEOPORTIOIf. 

SIMPLE  AND  COMPOUND  RATIO. 

195.  The  ratio  of  two  single  numbers  is  called  a  Simple MatiOf 
and  the  proportion  which  arises  from  the  equality  of  two  such 
ratios,  a  Simple  Proportion. 

If  the  terms  of  one  ratio  be  multipliod  by  the  terms  of  ano- 
tlier,  antecedent  by  antecedent,  and  consequent  by  consequent, 
the  ratio  of  the  products  is  called  a  Compound  Ratio.  Thus, 
if  the  two  ratios 

3     :     6     and     4     :     12 

be  multiplied  together,  Ave  shall  have  the  compound  ratio 

3x4     :     G  X  12,     or     12     :     72; 
in  which  the  ratio  is  equal  to  the  product  of  the  simple  ratios. 

A  proj)oriion  formed  from  the  equality  of  two  compound 
ratios,  or  from  the  equality  of  a  compound  ratio  and  a  simple 
ratio,  is  called  a  Compound  Proportion. 

196.    What  imrl  one  number  is  of  another. 

When  the  standard,  or  antecedent,  is  greater  than  the  num- 
ber which  it  measures,  the  ratio  is  a  proper  fraction,  and  is  such 
a  pai't  of  1,  as  the  number  measured  is  of  the  standard. 

Note. — The  standard  is  generally  preceded  by  the  word  o/,  and  in 
comparing  numbers,  may  be  named  second,  as  in  examples  7,  8,  9, 
10.  and  11,  but  it  must  always  be  used  as  a  divisor,  and  should  be 
placed  first  in  the  statement. 

1.  What  part  of  25  is  5  ?  that  is,  what  pai't  of  the  standard 
25,  is  5  ? 

-r^^  =1;     or     25     :     5     :  :     1     :     -^ ; 

that  is,  the  standard  is  to  the  number  measured  as  1  to  ^ ;  or, 
the  number  measured  is  one-fifth  of  the  standard. 


195.  What  is  a  simple  ratio  1  What  is  the  proportion  called  which 
cooies  from  the  equalily  of  two  simple  ratios  1  What  is  a  compound  ratio  I 
What  is  a  compound  jiroportion  1 

190.  When  the  .^t.iiidard  is  grcalrr  than  the  ronsrquont,  what  kind  of  a 
nnuilicr  is  the  ratio  ]  What  part  is  3  of  -1 !  C  of  12  ?  ^^'hal  part  of  4  is 
IC?    12  of  301 


RATIO    AND    PEOrOIiTION. 


VJ3 


2.  What  part  of  6  is  4  ? 

3.  What  part  of  10  is  5? 

4.  What  part  of  34  is  17  ? 

5  What  part  of  450  is  300? 

6.  What  part  of  96  is  16  ? 


7.  8  is  what  part  of  12? 

8.  IG  is  what  part  of  48  ? 

9.  18  is  what  part  of  90  ? 

10.  15  is  wliac  part  of  1G5  ? 

11.  9  is  what  par'  >f  11  ? 


SIMPLE  PROPORTION— OR,  SINGLE   RULE  OF  THREE. 

197.  Simple  Propoktion  is  an  expression  of  equality 
between  two  simple  ratios.  Hence,  a  simple  proportion  con- 
sists of  four  single  term,s,  in  which  the  ratio  of  the  first  to  the 
second  is  equal  to  the  ratio  of  the  third  to  the  fonrth.  If  three 
of  these  terms  are  known,  the  fourth  can  easily  be  found 
(Art.  193). 

198,  The  Rule  of  Three  explains  the  method  of  finding, 
from  three  given  numbers,  a  fourth,  to  wliich  the  third  shall 
bear  the  same  ratio  as  exists  between  the  first  and  second. 

1.  If  8  barrels  of  flour  cost  $5G,  what  will  9  barrels  cost, 
at  the  same  rate  ? 

Note. — We  shall  denote  the  vequired  terms  of  the  proportion  by 
the  letter  x. 


Analysis. — The  condition,  ''  at  the  same 
rate,"'  requires  that  the  quantity.  8  barrels 
of  flour,  have  the  same  ratio  to  the  quan- 
tity, 9  barrels,  as  $.56;  the  cost  of  8  barrels, 
to  X  dollars,  the  cost  of  9  barrels:  that  is, 

quantity  is  to  quantify,  as  cost  to  cost :   and, 

Since  the  product  of  the  two  extremes  is 
equal  to  the  product  of  the  two  means 
(Art.  193)  ;  we  have, 

8  X  0?   =   56  X  9  ; 


ST.\TEMENT. 

bar.    bar.       % 
8    :    9   : :    56 


OPERATION. 


$ 
X 


9 


X  =^  $63, 


197.  What    is  Simple   Proportion?     How  many  terms  are  employed? 
How  mnny  terme  must  be  Ivuown,  before  the  rest  can  be  found  ? 

198.  What  is  the  Kule  of  Three? 


194 


KATIO    AND    PROPORTION. 


and   if  8   times  x   is  equal   to    56  x  49,  x  must   be    equal    to   this 
product  divided  by  8  :  hence, 

The  fourth    term  is  equal  to  the  product  of    tht  second  and  third 
terms^  divided  by  the  fust. 

2.  If  36   dollars  will  buy  9  yards  of  cloth,  bow  many  yards, 
at  the  same  rate,  can  be  bought  for  $44  ? 


Analysis. — Thirty-six  dollars,  the  cost 
of  9  yards  of  cloth,  is  to  $44,  the  cost  of 
the  required  cloth,  as  9  yards  to  the  re- 
quired number  of  yards;  that  is,  cost  is  to 
cost  J  as  quantity  to  quantity. 

The  product  of  the  two  extremes  being 
equal  to  the  product  of  the  two  means,  we 
place  36  and  x  on  the  left  of  the  vertical 
line,  and  44  and  9,  on  the  right. 


STATEMENT. 

S       $  :  :  yd.  yd. 
36  :  44         9  :   X 


OPERATION. 


4 


t^ 


M 


11 


X  —    ilyd. 


199.     Hence,  we  have  the  following 

RULE. 

1.  Write  the  nvmher  which  is  of  the  same  kind  toith  the 
ansioer  for  the  third  term,  the  number  named  in  connection  with 
it  for  the  first  term,  and  the  remaining  number  for  the  second 
term. 

II.  Multiply  the  second  and  third  terms  together,  and  divide  the 
product  hy  tlie first  term:  or, 

Multiply  the  third  term  by  the  ratio  of  the  first  and  second. 

Notes. —  1.  If  the  fir.st  and  second  terms  have  diHeicnl  units,  ihoy 
must  be  reduced  to  the  same  unit. 

2.  If  the  third  term  is  a  compound  denominate  number,  it  must  be 
reduced  to  its  smallest  unit. 


109.    How  do  you  state  a  ([nestion  l>y  the  Rule  of  Three?     How  do  you 
find  the  foin-tli  term?     (live  the  entire  rule. 


EXAMPLES.  195 

3.  The  preparation  of  the  terms,  and  writing  them  in  their  proper 
places,  is  called  the  statement. 

4.  When  the   vertical   line  is   used,  the  unknown  term  is  always 
WTitten  at  the  left. 

EXAMPLES. 

1.  If  8  hats  cost  $24,  what  will  110  hats  cost,  at  the  same 
rate  ? 

2.  If  2  barrels  of  flour  cost  $15,  what  will  12  barrels  cost  ? 

3.  If  I  walk  168  miles  in  6  days,  how  far  should  I  walk,  at 
the  same  rate,  in  18  da}'s  ? 

4.  If  8^/;.  of  sugar  cost  $1,28,  how  much  will  13/^.  cost? 

5.  If  300  barrels  of  flour  cost  $2100,  what  will  125  barrels 
cost  ? 

6.  If  120  sheep  yield  330  pounds  of  wool,  how  many  pounds 
will  3G  sheep  yield  ? 

7.  If  80  yards  of  cloth  cost  $340,  what  will  650  yards  cost  ? 

8.  What  is  the  value  of  4:cwt.  of  sugar,  at  5  cents  a  pound  ? 

9.  If  6  gallons  of  molasses  cost  $1,95,  what  will  6  hogsheads 
cost? 

10.  If  16  men  consume  560  pounds  of  bread  in  a  month,  how 
much  would  40  men  consume  ? 

11.  If  a  man  travels  at  the  rate  of  630  miles  in  12  days 
how  far  will  he  travel  in  a  leap  year,  Sundays  excepted  ? 

12.  If  2  yards  of  cloth  cost  $3,25,  what   will  be  the  cost  of 
3  pieces,  each  containing  25  yards  ? 

13.  If  3  yards  of  cloth  cost  18s.  New  York  currency,  what 
will  36  yards  cost  ? 

14.  If  it  requires  eight  .shillings  and  four  pence  to  buy 
eight  ounces  of  laudanum,  how  many  ounces  can  be  purchased 
for  7s.  6(^.  ? 

15.  If  5 A.  IR.  16  P.  of  land,  cost  $150.5,  what  will  126A. 
'2R.  20P.  cost  ? 

16.  If  Vdcwt.  -Iqr.  of  sugar  cost  $129,93,  what  will  be  the 
cost  of  ^cwt.  ? 

17.  The  clothing  of  a  regiment  of  750  men  cost  £2834  os.: 
what  will  it  cost  to  cloth    a  body  of  10500  men  ? 


196  SIMPLE    PROPORTION. 

18.  If  dyd.   2qr.  of  cloth  cost  $15,75,  how  much  will  ^yd, 
oqr.  of  the  same  cloth  cost  ? 

19.  If  .5  of  a  house  cost  $201.5,  what  would  .95  cost  ? 

20.  What  will  26.25  bushels  of  wheat   cost,  if  o.o  bushels 
cost  48.40  ? 

21.  If  the  transportation  of  2.5  tons  of  goods  2.8  miles  costs 
$1,80,  what  is  that  per  cwt.'^ 

22.  If  f  of  a  yard  of  cloth  cost  $2,1 6,  what  will  I  of  a  yard 
cost  ? 

23.  If  ^  of  an  ounce  cost  $11,  what  will  l^oz.  cost  ? 

24.  What  is  the  cost  of  lG|/&.  of  sugar,  if  14p.  cost  $lf  ? 

25.  If  $19}  will  buy  14^  yards  of  cloth,  how  much  will 
39|  yards  cost  ? 

26.  If  |-  of  a  barrel  of  cider  cost  -jj  of  a  dollar,  what  will 
li  of  a  barrel  cost  ? 

27.  If  T^  of  a  ship  cost  $2880,  what  will  i§  of  her  cost? 

28.  What  will  116i  yards  of  cloth  cost,  if  462  yards  cost 
$150,66? 

29.  If  7-/j-  barrels  of  fish  cost  $311  what  will  32^  barrels 
cost  ? 

30.  How  much  wheat  can  be  bought  for  $9G|-,  if  ^hu.  Ipk. 
cost  $1,93 1  ? 

31.  If '^^  of  a  yard  of  cloth  cost  $1|,  what  will  71  yards  cost? 

32.  What  will  be  the  cost  of  37.05  square  yards  of  pavement, 
if  47.5  yards  cost  $72.25  ? 

33.  If  3  paces  or  common  steps  be  equal  to  2  yards,  how 
many  yards  will  160  paces  make  ? 

34.  If  a  person  pays  half  a  guinea  a  week  for  his  board,  how 
long  can  he  board  i'or  £21  ? 

35.  If  12  dozen  copies  of  a  certain  book  cost  $54.72,  what 
will  297  copies  cost  at  the  same  rate  ? 

36.  If  $3618  worth  of  provisions  will  subsist  an  armv  of 
9000  men  for  90  days,  if  tiie  army  be  increased  by  4500  men, 
how  much  would  last  them  the  same  time  ? 

37.  A  grocer  bought  a /</<rf.  of  nun  ibr   80   cents    a    gallon, 


EXAMPLES.  197 

and  after  adding  water  pold  it  for  GO  cents  a  gallon,  when  lie 
found  that  the  selling  and  buying  prices  were  proportional  to  the 
original  quantity  and  the  mixture  :  how  much  water  did  he  add  ? 

38.  A  man  failing  in  business,  pays  GO  cents  for  every  dollar 
wliicli  he  owes  ;  he  owes  A  83570,  and  B  '"5^1875  ;  how  much 
does  he  pay  each  ? 

39.  A  bankrupt's  effects  amount  to  82328,75,  his  debts 
amount  to  $3726  :  what  will  his  creditors  receive  on  a  dollar  ? 

40.  If  a  person  drinks  80  bottles  of  wine  in  3  months  of  30 
days  each,  how  much  does  he  drink  in  a  week  ? 

41.  If  4|-  yards  of  cloth  cost  14s.  8d.,  New  York  currency, 
what  will  401  yards  cost  ? 

42.  If  a  grocer  use  a  false  balance  giving  only  lA-oz.  for  a 
pound,  how  much  will  154^lbs.  of  just  weight  give,  when  weighed 
by  the  false  balance  ? 

43.  If  a  dealer  in  liquors  use  a  gallon  measure  which  is  too 
small  by  ^  pint,  what  will  be  the  true  measure  of  100  of  the 
false  gallons  ? 

44.  After  A  has  travelled  96  miles  on  a  journey,  B  sets  out 
to  overtake  him,  and  travels  23  miles  as  often  as  A  travels  19 
miles  :  how  far  will  B  travel  before  he  overtakes  him  ? 

45.  A  person  owning  y  of  a  coal  mine,  sold  j  of  his  share  for 
$9345  ;  what  was  the  value  of  the  whole  mine  ? 

4G.  At  wiiat  time  between  6  and  7  o'clock,  will  the  hour  and 
minute  hands  of  a  clock  be  exactly  together? 

47.  If  a  staff  5  feet  long  casts  a  shadow  7  feet,  what  is  the 
height  of  a  steeple,  whose  shadow  is  196  feet  at  the  same  time 
of  dav  ? 

48.  Two  persons  are  279  miles  apart,  start  at  the  same  time 
and  travel  toward  each  other.  A  goes  5  miles  an  hour,  and  B 
4  miles :  how  many  miles  must  each  travel  before  they  meet  ? 

49.  A  can  do  a  piece  of  work  in  3  days,  B  in  4  days,  and  C 
tn  G  days  :  in  what  time  will  they  all  do  it,  working  together  ? 

50.  A  can  build  a  wall  in  15  days,  but  with  the  assistance  of 
C,  he  can  do  it  in  9  days  :  in  wlmt  time  can  C  do  it  alone .' 


198  IXVKltSE    PBorOUTIOX. 

-* 
51.  A  and  B  take  a  job  for  which  they  are  to  receive  $1 65.75  ; 

A  M'orks  himself  and   emi)loys  7  hands  ;   B  does  the  same  and 

employs  6  hands  :  what  should  eacl^receive  ? 

o2.  A  watch,  which  is  10  minutes  too  fast  at  12  o'clock,  on  j\Ion- 

day,  gains  Smin.  lOsec.  per  day :  what  will  be  the  time  by  the  watch 

at  a  quarter  past  ten  in  the  morning  of  the  following  Saturday  ? 

53.  There  are  two  clocks,  one  of  which  gains  10  miimtes, 
and  the  other  loses  7^  minutes  every  24  hours.  They  are  to- 
gether at  noon  on  Tuesday  :  what  will  be  the  difference  of  their 
times  at  6  o'clock  on  Friday  morning  ? 

54.  If  15  men  can  be  boarded  1  week  for  846,25,  what  will 
it  cost  to  board  5  men  and  6  boys,  the  same  time,  the  boys 
being  boarded  at  half  price  ? 

55.  Two  persons,  A  and  B,  are  on  the  opposite  sides  of  a 
wood,  which  is  536  yards  in  circumference  ;  they  begin  to  travel 
in  the  same  direction  at  the  same  moment ;  A  goes  at  the  rate 
of  11  yards  per  minute,  and  B  at  the  rate  of  34  yards  in  3  min- 
utes :  how  many  times  must  A  go  round  the  wood  before  he  is 
overtaken  by  B  ? 

CAUSE  AND  EFFECT. 

200.  Whatever  produces  Effects,  as  men  at  work,  animals 
eating,  time,  goods  purchased  or  sold,  money  lent  producing 
interest,  and  the  like,  may  be  regarded  as  Causes. 

Causes  are  of  two  kinds,  simple  and  compound  : 

A  Simple  Cause  has  but  a  single  element,  as  men  at  work, 

a  portion  of  time,  goods  [)urchased  or  sold,  and  the  like. 

A  Compound  Cause  is  made  up  of  two  or   more    simple 

elements,  such  as  men  at  work  taken   in  connection  with  time, 

and  the  like. 

201.  The  results  of  causes,  as  work  done,  provisions  con- 
sumed, money  paid,  cost  of  goods,  and  the  like,  maybe  regarded 
as  effects. 

'21)0.  ^V lint  are  causes?  How  many  kinds  of  causes  arc  there?  AVliat 
is  a  simple  cause?     What  is  a  compouiul  cause  ? 

201.  What  are  effects  V  What  is  a  simple  clVect?  What  is  a  compouud 
effect? 


mvp:KsE  PKoi'ouTiox.  199 

A  Simple  Effect  is  one  which  has  but  a  single  element. 
A  Compound  Effect  is  one  which  arises  from   the   com- 
bination of  two  or  more  elements. 

202.  Causes  which  are  of  the  same  kind,  that  is,  which  can 
be  reduced  to  the  same  unit,  may  be  compared  with  each  other; 
and  effects  which  are  of  the  same  kind,  may  likewise  be  com- 
pared with  each  other.  From  the  nature  of  causes  and  effects, 
we  know  that, 

1st  Cause     :     2d  Cause     :  :     1st  Effect     :     2d  Effect ; 
and,  1st  Etfect     :     2d  Effect     :  :     1st  Cause     :     2d  Cause. 

203.  Simple  causes  and  simple  effects  give  rise  to  simple  ratios. 
Compound  causes  or  compound  effects  give  rise  to  compound 

ratios. 

204.  All  questions  involving  simple  ratios  are  classed  under 
Simple  Proportion;  and  all  questions  involving  compound 
ratios,  either  under  Inverse  or  Compound  Proportion. 

INVERSE   PROPORTION. 

205.  It  often  happens,  that  two  numbers  which  are  compared 
with  each  other,  undergo,  or  may  undergo,  certain  changes  of 
value,  in  which  case  they  represent  variable  and  not  Jived  quan- 
tities. Thus,  when  we  say,  that  the  amount  of  work  done,  in  a 
single  day,  will  be  proportional  to  the  number  of  men  em- 
ployed, we  mean,  that  if  we  increase  the  number  of  men,  the 
amount  of  work  done  will  also  be  increased';  or,  if  we  diminish 
the  number  of  men  employed,  the  work  done  will  also  be 
diminished.     This  is  called  Direct  Proportion. 

If  we   say  that  a  barrel  of  flour  will  serve  12  men  a  certain 

202.  What  causes  are  of  the  same  kind  ?  What  causes  mar  be  com- 
pared with  each  other  ?  What  do  we  infer  from  the  nature  of  causes  and 
eJfects  ? 

203.  Wliat  ratios  arise  from  simple  causes  and  simple  effects?  From 
what  do  compound  ratios  arise  ? 

204.  What  questions  fall  under  simple  proportion  ?  Questions  involving 
compound  ratios  give  rise  to  what  kinds  of  proportion  ? 

205.  When  are  two  numbers  directly  proportional  ?  When  are  two 
numbers  inversely  proportional  ?     Does  their  product  then  vary  ? 


200  IKVKRSE    PROPOKTIOX. 

time,  and  ask  hoAv  long  it  will  serve  24  men,  there  is  a  certain 
relation  between  the  number  of  men  and  time  ;  but  that  relation 
is  such  that  the  time  will  decrease,  if  the  number  of  men  is 
mcreased,  and  will  increase,  if  the  number  of  men  is  decreased 
This  is  called,  Inverse  Proportion  ;  hence, 

1.  Two  numbers  are  directly  proportional,  tuhen  they  increase 
or  decrease  together  ;  in  which  case  their  ratio  is  always  the  same. 

2.  Two  numbers  are  inversely  or  reciprocally  projoortional, 
when  one  increases  as  the  other  decreases  ;  in  which  case  their 
product  in  ahoays  the  same. 

Note. — This  is  sometimes  called  Reciprocal  Proportion. 

206.  If  we  refer  to  the  numeration  table  of  integral  and 
decimal  numbers  (Art.  14G),  we  see  that  the  unit  of  the  first 
place,  at  the  left  of  1,  is  1  ten  ;  that  is,  a  number  ten  times  as 
great  as  1.  The  unit  of  the  first  decimal  place  at  the  right,  is  1 
tenth,  a  number  only  one-tenth  of  1.  The  unit  of  the  second 
place,  at  the  left,  is  one  hundred  times  as  great  as  1  ;  while  the 
unit  of  the  second  place,  at  the  right,  is  only  one  hundredth  of 
1 ;  and  similai'ly  for  all  other  corresponding  places  ;  hence. 

The  units  of  jylace,  taken  at  equal  distances  from  the  unit  1, 
are  inversely  2}roportional. 

207.  1- — The  floor  of  a  room  is  20  feet  long  :  what  must  be 
its  breadth,  in  order  that  it  may  contain  360  square  feet  ? 

Analysis. — The  length  of  the  floor,  multi-  opehation. 

plied  by  its   breadth,  will   give  the  area  or        3()0 

.     /      .  ..  ■  J-    •;!    1  1      .1  --  =  18//.  breadth, 

contents;    hence,  the    area,    divided  by  the         20  *' 

length,  will  give  the  breadtli. 

Note. — When  the  area  or  contents  of  a  room  are  known,  or  given, 


206.  What  relation  exists  between  the  units  of  place  in  the  integral  and 
decimal  numeration  table  ?     Give  an  example  ? 

207.  What  arc  tlie  contents  of  a  floor  equal  to?  What  is  the  breadth 
equal  to?  When  tlio  contents  of  a  floor  are  given,  in  wbat  proportion  is 
the  length  to  the  breadth?  If  two  numbers  are  inversely  proportional, 
what  is  either  equal  to  ? 


INVERSE    PROPORTION. 


201 


the  length  of  the  room  is  inversely  proportional  to  its  breadth  (Art. 
205).  Il'  the  length  of  a  room,  which  is  to  contain  a  given  area,  bo 
increased  any  number  of  times,  the  breadth  of  the  room  mu.st  be  dimin- 
ished just  as  many  times.  If  the  length  be  divided  by  any  number, 
the  breadth  must  be  mulliplied  by  the  same  number.  Thus,  two 
rooms,  one  40  feet  long  and  9  feet  wide,  the  other,  10  feet  oneway 
and  3(5  feet  the  other,  have  the  same  area,  viz.,  360  square  feet. 
Hence,  when  two  numbers  ar§  inversely  proportional,     * 

Either  is  equal  to  their  product  divided  by  the  other. 

208.  We  may  regard  the  length  of  a  room  as  one  cause  of 
its  contents,  and  its  breadth  as  another  cause  of  its  contents; 
foi*,  the  contents  being  equal  to  the  product  of  the  length  and 
breadth,  is  the  effect  of  them  both. 

In  the  case  of  the  tv»'o  rooms,  one  40  feet  long  and  9  feet 
wide,  and  the  other  10  feet  by  36,  the  effects  are  the  same. 
The  causes  are  compound,  each  being  composed  of  two  elements 
(Art.  200) ;  and  since  the  effects  ai-e  equal  the  causes  are  equal 
(Art.  202)  ;  hence, 

Whtn  the  causes  are  equcd,  the  elements  are  inversely  pro- 
portional. 

1.  If  3G  men,  in  12  days,  can  do  a  certain  work,  in  what  time 
will  48  men  do  the  same  work  ? 


Analysis. — The  first  cause  is 
compounded  of  36  men  and  12 
days,  and  Ls  equal  to  36  x  12  =  432, 
the  number  of  days  it. would  require 
1  man  to  do  tlie  work. 

The  second  cause  is  compounded 
of  48  men  and  the  number  of  days 
it  would  require  them  to  do  the 
Bame  work,  and  is  equal  lo  48  x  x. 

Bat  since  the  effects  are  the  same, 
viz.,  the  work  done,  the  causes  must 
be  equal ;  hence,  the  products  of  the 
elements  of  the  causes  are  equal. 
Tlierefore,  in  the  solution  of  all 
like  examples, 


STATEMENT 

meji. 

men.  ' 

36 

:     48 

days. 
12 

days. 

X 

>      ; 

432 

:     AS  XX    : 

OPERATION. 

X 

9 

Ans.  X  =  9  days. 


202  EXAMPLES. 

Write  the  elements  of  the  cause  containing  the  unknown  ele- 
ment on  the  left  of  the  vertical  line  for  a  divisor,  and  the  elcmenti 
of  the  oilier  cause  on  the  right  for  a  dividend. 

Notes. — 1.  Since  the  effects  are  equal,  they  may  each  be  denoted 
by  1  ;  hence,  the  causes  are  to  each  other  as  1  to  1 . 

2.  It  is  evident,  that  in  this  class  of  questions  the  elements  of  the 
causes  are  inversely  proportional ;  and  hence,  such  questions  have 
generally  been  arranged  under  the  head  of  '•  Rule  of  Three  Inverse." 

EXAMPLES. 

1.  If  3|  yards  of  cloth  will  make  a  coat  and  vest,  -when  the 
cloth  is  1^  yards  wide,  how  much  cloth  will  be  needed  which  is 
•I  yards  in  width  ? 

2.  If  I  have  a  piece  of  land  16f  rods  long  and  3^  rods 
wide,  what  will  be  the  length  of  another  piece  that  is  7  rods 
wide  and  contains  an  equal  area  ? 

3.  How  many  yards  of  carpeting  that  is  three-fourths  of 
a  yard  wide,  will  carpet  a  room  36  feet  long  and  30  feet  in 
breadth  ? 

4.  If  a  man  can  perform  a  journey  in  8  days,  walking  9 
hours  a  day,  how  many  days  will  it  require  if  he  walks  10  hours 
a  day  ? 

5.  If  a  family  of  15  persons  have  provisions  for  8  months, 
by  how  many  must  the  family  be  diminished  that  the  provisions 
may  last  2  years  ? 

6.  A  garrison  of  4  GOO  men  have  provision  for  G  months  :  to 
what  number  must  the  garrison  be  diminished  that  the  provi- 
sions may  last  2  years  and  G  months  ? 

7.  A  certain  amount  of  provisions  will  subsist  an  army  of 
9000  men  for  90  days  :  if  the  army  be  increased  by  6000,  how 
long  will  the  same  provisions  subsist  it  ? 


208.  M'hat  may  we  regard  as  causing  or  producing  the  contents  of  a 
room  ?  When  two  causes  arc  equal,  how  are  the  elements  1  Note. — It 
the  cifects  are  equal,  by  what  may  they  be  denoted  1 


EXAMPLES.  203 

8.  If  G  men  and  3  boys  can  do  a  piece  of  work  in  330  days, 
how  long  will  it  take  9  men  and  4  boys  to  do  the  same  work, 
under  the  supposition  that  each  boy  does  half  as  much  as  a 
man  ? 

9.  Four  thousand  soldiers  were  supplied  with  bread  for  24 
weeks,  each  man  to  receive  liios.  per  day;  but,  by  some  acci- 
dent, 210  barrels  containing  200/6.  each  were  spoiled:  what 
must  each  man  receive  in  order  that  the  remainder  may  last  the 
same  time  ? 

10.  Suppose  4000  soldiers  after  losing  210  barrels  of  bread 
each  containing  2001b.,  were  to  subsist  on  loos,  each  a  day  for 
24  weeks  ;  had  none  been  lost  they  would  have  received  14:0Z. 
a  day  :  what  was  the  weight  of  the  whole,  and  how  much  did 
they  receive  ? 

11.  Let  us  now  suppose  4000  soldiers  to  lose  one-fourteenth 
of  their  bread,  then  to  receive  looz.  each  a  day  for  24  weeks  : 
what  was  the  whole  weight  of  their  bread  including  the  lost,  and 
how  much  would  each  have  received  per  day  had  none  been 
spoiled  ? 

12.  If  4  men  can  do  a  piece  of  work  in  80  days,  how  many 
days  will  16  men  require  to  do  the  same  Avork? 

13.  If  21  pioneers  make  a  trench  in  18  days,  how  many  days 
will  7  men  require  to  make  a  similar  trench  ? 

14.  A  certain  piece  of  grass  was  to  be  mowed  by  20  men  in 
6  days ;  one-half  the  workmen  being  called  away,  it  is  required 
to  find  in  what  time  the  remainder  will  complete  the  work? 

15.  If  a  field  of  grain  be  cut  by  10  men  in  12  days,  in  how 
many  days  would  it  be  cut  by  20  men  ? 

16.  If  90  barrels  of  flour  will  subsist  100  men  for  120  days, 
how  long  will  it  subsist  75  ? 

17.  If  a  traveller  perform  a  journey  in  35.5  days,  when  the 
days  are  13.566  hours  long,  in  how  many  days  of  11.9  hours 
would  he  perform  the  same  journey  ? 

18.  If  50  persons  consume  600  bushels  of  wheat  in  a  year, 
how  long  would  it  last  5  persons  ? 

19.  A  certain  work  can   be  done  in  12  days,  by  working  4 


20'i:  INVEKSE    PEOPOKTION. 

hours  each  day  :  how  many  days  would  it  require  to  do  the  same 
work  by  working  9  hours  a  day  ? 

20.  If  120  men  can  build  ^  mile  of  wall  in  15^  days,  how 
many  men  would  it  require  to  build  the  same  wall  in  40|  days  ? 

21.  A  garrison  of  oGOO  men  has  just  bread  enough  to  allow 
24:02.  a  day  to  each  man  for  34  days  ;  but  a  siege  coming  on, 
the  garrison  was  reinforced  to  the  number  of  4800  men.  How 
many  ounces  of  bread  a  day  must  each  man  be  allowed,  to  hold 
out  45  days  against  the  enemy  ? 

22.  If  3  horses  or  5  colts  eat  a  certain  quantity  of  oats  in  40 
days,  in  what  time  will  7  horses  and  3  colts  consume  the  same 
quantity  ? 

23.  If  a  person  can  perfoi'm  a  journey  in  24  days  of  10^ 
hours  each,  in  what  time  can  he  perform  the  same  journey, 
when  the  days  are  12i  hours  long  ? 

24.  A  piece  of  land  40  rods  long  and  4  rods  wide,  is  equiva- 
lent to  an  acre :  what  is  the  breadth  of  a  piece  15  rods  long 
that  is  equivalent  to  an  acre  ? 

25.  If  a  person  travelling  1 2  hours  a  day  finish  one  half  of  a 
journey-  in  10  days,  in  what  time  will  he  finish  the  remaining 
half,  travelling  9  hours  a  day  ? 

26.  How  many  pounds  weight  can  be  carried  20  miles  for  the 
same  money  that  4-^  hundred  weight  can  be  carried  36  miles? 

27.  If  20  men  can  perform  a  piece  of  work  in  12  days, 
working  9  hours  a  day,  how  many  men  will  accomplish  the 
same  work  in  one  half  the  time,  working  10  hours  a  day? 

28.  If  72  horses  eat  a  certain  quantity  of  hay  in  7^  weeks, 
how  many  horses  will  consume  the  same  in  90  weeks  ? 

29.  Bought  5000  planks,  15  feet  long  and  2^  inches  thick; 
how  many  planks  are  they  equivalent  to,  of  12^2-  feet  long  and 
1^-  inches  thick  ? 

30.  If  12  pieces  of  cannon,  eighteen  pounders,  can  batter 
down  a  castle  in  3  hours,  in  what  time  would  nine  twenty-four 
])ounders  batter  down  the  same  castle,  both  pieces  of  cannon 
being  fired  the  same  number  of  times,  and  their  balls  flying 
with  the  same  velocity? 


COMPOUND    PROPOKTION. 


205 


COMPOUND  PROPORTION. 

209.  Compound  Proportion  is  a  comparison  of  compound 
ratios  wlieii  the  terms  are  unequal. 

It  embraces  that  class  of  questions  in  which  the  causes  arc 
compound,  or  in  which  the  effects  are  compound.  In  this  class 
of  questions,  either  a  cause  or  a  single  element  of  a  cause,  may 
be  required;  or  an  effect,  or  a  single  element  of  an  effect  may 
be  required. 

1.  If  8  men  in  12  days  can  build  80  rods  of  wall,  how  much 
will  6  men  do  in  18  days? 


or 


STATEMENT. 

Cause     : 

2d  Cause     :  :     1st  Efl^ct 

:     2d  Effect. 

12;   • 

is}       -  '0 

:     X 

12x8  : 

18x6        :  :     80 

:     X 

Analysis. —  In  this  example  the  second 
effeet  is  required,  and  the  statement  may  bs 
read  thus  : 

If  8  men  in  12  days  can  build  80  rods  of 
wall,  6  men   in   18   dajs    will   build    how 
many  (or  x)  rods  of  wall  ? 


OPERATTON. 


t 


If. 

X 


1$ 


$0  10 


Ans.  X  =z  90  rods. 


2.  If  a  family  of  12  persons,  in  8  months,  expend  $864,  hov/ 
many  months  will  $900  serve  a  family  of  20  persons  ? 


or. 


STATEMENT. 

1}  ^ 

^.r}          '      ''           ^^^^ 

$900. 

12x8    : 

20xx     :     :         $864     : 

$900. 

209.  What  is  compound  proportion  1    What  questions  docs  it  eml)race' 
Wliat  i.9  ahvays  required  ^ 


20G 


COMPOUND    PROI'OKTIUN. 


Analysis. — In  this  example,  an  element 
of  the  second  cause  is  required,  viz..  the  num- 
ber of  months  which  the  money  will  last  20 
men.     The  question  is  thus  stated  : 

If  12  persons,  in  8  months,  expend  $864, 
20  persons  in  how  many  (or  x)  mouths  will 
expend  $900  ? 


OPER.iTION. 


t% 


X 


n 


$u 


$     5 


000 


Alls.  X  :=  5  months. 


o.  If  24  men,  in  G  days,  working  7  hours  a  clay,  can  buiJd  a 
wall  115  feet  long  3  feet  thick  and  4  feet  high,  how  long  a  wall 
can  36  men  build  in  12  days,  working  14  hours  a  day,  if  the 
wall  is  4  feet  thick  and  5  feet  in  height  ? 


STATEMENT. 


115 


or,       24x6x7    :    36x12x14 


115x3x4    :    .rx4x5. 


Analysis. — In  this  example,  an  element 
of  the  second  effect  is  required,  viz..  the 
length  of  the  wall ;  and  the  question  may 
be  thus  stated : 

If  24  men,  in  6  days,  working  7  hours  a 
day,  can  build  a  wall  115  feet  long,  3  feet 
thick,  and  4  feet  high,  36  men  in  12  days, 
working  14  hours  a  day,  will  build  a  wall 
how  many  (or  x)  feet  long,  4  feet  thick  and 
5  feet  high? 


OPERATION. 


u 

0 

t 

X 

Ans. 


$6 

n$  2^ 

3 

.r  =  450/V. 


210.  ITcncc,  we  have  the  following 

Rule. — I.  Arrange  the  terms  in  the  statement  so  that  the 
causes  shall  comjiose  one  cotq^let,  and  the  effects  the  other,  jii^li^'"-!? 
X  in  the  2'>l'ice  of  the  required  element. 


110.  Give  the  rule  for  stating  the  question  and  finding  the  unknown 


part 


EXAMPLES.  207 

II.  If -s.  fall  in  either  extreme,  make  the  product  of  the  means 
a  dividend,  and  the  product  of  the  extremes,  omitting  x,  a  dieisor  ; 
if  yi  fill  <n  cither  mean,  make  the  product  of  the  extremes  a  divi- 
dend, and  the  product  of  the  means,  omittinf/  x,  a  divisor. 

EXAMPLES. 

1.  If  2  men  can  dig  125  rods  of  ditch  in  75  days,  in  Low 
many  days  can  18  men  dig  243  rods  ? 

2.  If  400  soldiers  consume  5  barrels  of  flour  in  12  days,  how 
many  soldiers  will  consume  15  barrels  in  2  days  ? 

3.  If  a  person  can  travel  120  miles  in  12  days  of  8  hours 
each,  how  far  will  he  be  able  to  travel  in  15  days  of  10  hours 
each  ? 

4.  If  a  pasture  of  16  acres  will  feed  G  horses  for  4  months, 
how  many  acres  will  feed  12  horses  for  9  months  ? 

5.  If  60  bushels  of  oats  wiU  feed  24  horses  40  days,  how  long 
will  30  bushels  feed  48  horses  ? 

6.  If  82  men  build  a  wall  36  feet  long,  8  feet  high,  and  4 
feet  thick,  in  4  days,  in  what  time  will  48  men  build  a  wall  864 
feet  Ions;,  6  feet  biijh,  and  3  feet  wide  ? 

7.  If  the  freisrht  of  80  tierces  of  sugar,  each  weighinoj  3i 
hundred  weight,  for  150  miles,  cost  $84,  what  must  be  paid  for 
the  freight  of  30  hogsheads  of  sugar,  each  weighing  12  hundred 
weight,  for  50  miles  ? 

8.  A  family  consisting  of  6  persons,  usually  drink  15.6  gal- 
lons of  beer  in  a  week:  how  much  will  they  drink  in  12.5 
weeks,  if  the  number  be  increased  to  9  ? 

9.  If  12  tailors  in  7  days  can  finish  14  suits  of  clothes,  how 
many  tailors  in  19  days  can  finish  the  clothes  of  a  regiment  of 
494  men  ? 

10.  If  a  garrison  of  3600  men  eat  a  certain  quantity  of  bread 
in  35  days,  at  24  ounces  per  day  to  each  man,  how  many  men,  at 
the  rate  of  14  ounces  per  day,  will  eat  twice  as  much  in  45  days  ? 

11.  A  company  of  100  men  drank  £20  worth  of  wine  at  2.?. 
6(/.  per  bottle  :  how  many  men,  at  the  same  rate,  will  £7  worth 
supply,  when  wine  is  worth  l6'.  9t/.  per  bottle  ? 


208  OOilPOUND    PKOPOKTION. 

12.  If  the  Avages  of  13  men  for  74-  days,  be  $149,76,  -what 
will  be  the  wages  of  20  men  for  15  i  days  ? 

I'd.  If  a  footman  travel  20-4  miles  in  Gg-  days  of  121-  hours 
each,  in  how  many  days  of  10|-  hours  each  will  he  travel  120^ 
miles  ? 

14.  If  120  men  in  3  days,  of  12  hours  each,  can  dig  a 
trench  of  30  yards  long,  2  feet  broad,  and  4  feet  deep,  how 
many  men  would  be  required  to  dig  a  trench  50  yards  long, 
6  feet  deep,  and  li  yards  broad,  in  9  days  of  15  hours 
each  ? 

15.  If  40  men  can  perform  a  piece  of  work  in  12  days,  how 
many  men  will  perform  another  piece  of  work  three  times  as 
large,  in  one-fifth  part  of  the  time  ? 

16.  A  person  having  a  journey  of  500  miles  to  perform, 
walks  200  miles  in  8  days,  walking  12  hours  a  day:  in  how 
many  days,  walking  10  hours  a  day,  will  he  complete  the 
remainder  of  the  journey? 

17.  If  1000  men,  besieged  in  a  town,  with  provisions  for  28 
days,  at  the  rate  of  18  ounces  per  day  for  each  man,  be  rein- 
forced by  600  men,  how  many  ounces  a  day  must  each  man 
have  that  the  provisions  may  last  them  42  days  ? 

18.  If  a  bar  of  iron  5/f.  long,  2li>i.  wide,  and  l^-in.  thick, 
weigh  Ablbs.  how  much  will  a  bar  of  the  same  mcU\l  wei/di  that 
is  Ifl.  long,  dill,  wide,  and  2i/».  thick  ? 

19.  If  5  compositors  in  16  days,  working  14  hours  a  day, 
can  compose  20  sheets  of  24  pages  each,  50  lines  in  a  page,  and 
40  letters  in  a  line,  in  how  many  days,  working  7  liou-s  a  day, 
can  10  compositors  compose  40  sheets  of  16  pages  in  a  sheet, 
60  lines  in  a  page,  and  50  letters  in  a  line  ? 

20.  Fifty  thousand  bricks  are  to  be  removed  a  given  distance 
in  10  days.  Twelve  horses  can  remove  18000  in  6  ^%ys:  how 
many  horses  can  remove  the  remainder  in  4  days? 

21.  If  3  men,  working  10  hours  a  day,  can  plant  }■  field  150 
rods  by  240  rods,  in  5  days,  how  many  men,  working  12  hours 
a  day,  can  plant  a  field  measuring  192  rods  by  30t>  vod^^,  in  4 
days  ? 


PAI^TNEUSITTP.  200 

22.  If  248  men,  in  5  J,  days  of  11  hours  each,  dig  a  trench  of 
7  degrees  of  hardness,  232^-  yards  long,  of  wide,  and  2^  deep, 
in  how  many  days,  of  9  hours  long,  will  24  men  dig  a  trench  of 
4  degrees  of  hardness,  337-^  yards  long,  5|  wide,  and  31  deep  ? 

PARTNERSHIP. 

211.  Partxership  is  the  joining  together  of  two  or  more 
persons  in  trade,  with  an  agreement  to  share  the  profits  or 
losses. 

Partners  are  those  who  are  joined  together  in  carrying  on 
business. 

Capital,  is  the  amount  of  money  employed. 
Dividend    is  the  gain  or  profit : 
Loss  is  the  opposite  of  jirofit. 

212.  The  Capital  or  Stock  is  a  cause  of  the  entire  profit : 
Each  man's  capital  is  the  cause  of  his  profit : 

Tlie  entire  profit  or  loss  is  the  effect  of  the  cause  or  capital : 
Each  man's  profit  or  loss  is  the  effect  of  his  capital :  hence. 
Whole  Stock  :  Each  man's  Stock 
:  :  Whole  profit  or  loss  :  Each  man's  profit  or  loss. 
1.  Mr.  Jones  and  Mr.  Wilson  are   partners  in  trade:  Mr. 
Jones  puts  in,  as  capital,  $1250,  and  Mr.  Wilson,  $750  :  at  the 
end  of  a   year  there  is  a  profit  of  $720  :  what  is  the  share  of 
each? 

STATEIVIENT.  OPERATION. 


2000  :  1250  ::  720  :  Jones' share.     ^^^^^ 

X 


2000  :     750  ::  720  :  Wilson's  share. 


^ 


X  —  $450  Ans. 
15 

X 


liv- 


n^ 


X  =  270. 


211.  What  is  a  partnership?     What  are  partners'?     What  is  capital  oi 
<ock  ?     What  is  dividend  ?     What  is  loss  ? 


210  PARTNEKSniP. 

Hence,  the  following 

Rule. — As  f/ie  iv/wie  stock  is  to  each  tnan^s  share,  so  is  the 
whole  gain  or  loss  to  each  man^s  share  of  the  gain  or  loss. 

EXAJIPLES. 

1.  A,  B,  and  C,  entered  into  partnership  with  a  capital  of 
$7500,  of  which  A  put  in  $2500,  B  put  in  $3000,  and  C  put 
in  the  remainder  ;  at  the  end  of  the  year  their  gain  was  §3000  : 
what  was  each  one's  share  ? 

2.  C  and  D  have  a  joint  stock  of  $4200,  of  which  A  owns 
$3600,  and  B,  $600  :  they  gain  in  one  j-ear,  §2000  :  what  is 
each  one's  share  of  the  profits  ? 

3.  A,  B,  C,  and  D,  have  §40,000  in  trade,  each  an  equal 
shai'c  ;  at  the  end  of  six  months  their  profits  amount  to  §16000  : 
what  is  each  one's  share,  allowing  A  to  recfeive  $50,  and  D,  $30, 
out  of  the  profits,  for  extra  services  ? 

4.  Five  persons  have  to  share  between  them  an  estate  of 
$20000  ;  A  is  to  have  one-fourth,  B  one-eighth,  C  one-sixth, 
D  one-eighth,  and  E  what  is  left :  what  will  be  the  share  of 
each  ? 

5.  Three  merchants  loaded  a  vessel  with  flour  ;  A  loaded  500 
barrels,  B,  700  barrels,  and  C,  1000  barrels ;  in  a  storm  at  sea, 
it  became  necessary  to  throw  overboard  440  barrels  ;  what  was 
each  one's  share  of  the  loss  ? 

6.  A  man  bequeathed  his  estate  to  his  four  sons,  in  the  fol- 
lowing manner,  viz.  :  to  his  first,  §5000,  to  his  second,  §4500, 
to  his  third,  $4500,  and  to  his  fourth,  $4000.  But  on  settling 
the  estate,  it  was  found  that  after  paying  the  debts  and  expenses, 
only  $12000  remained  to  be  divided  :  how  much  should  each 
receive  ? 

7.  A  widow  and  her  two  sons  receive  a  legacy  of  $1500,  of 
which  the  widow  is  to  have  -^,  and  the  sons  each  i.  But  the 
oldest  son  dying,  the  whole  is  to  be  divided  in  the  saire  propor- 
tion between  the  mother  and  youngest  son :  whai  will  each 
receive .'' 

8.  Four  persons  engage  jointly  in  a  land  speculation  :  D  puts 


COMPOUND    I'AETNEKSHIP.  211 

in  $5499  capital.  Tiiey  gain  $15000,  of  which  A  takes  $4320,50, 
B,  $5245,75,  and  C,  $3600,75  :  how  much  capital  did  A,  B, 
and  C  put  in,  and  what  is  D's  sliare  of  the  gain  ? 

9.  A  steam-mill,  valued  at  $4300,  was  entirely  destroyed  by  fire. 
A  owned  i  of  it,  B  ^,  and  C  the  remainder  ;  supposing  it  to  have 
been  insured  for  $2500,  what  Avas  each  one's  share  of  the  loss  ? 

10.  A  copartnership  is  formed  with  a  joint  capital  of  $10970. 
A  puts  in  $5  as  often  as  B  puts  in  $7,  and  as  often  as  C  puts 
in  $8  ;  their  annual  gain  is  equal  to  C's  stock  :  what  is  each 
person's  stock  and  gain  ? 

11.  A  man  failing  in  business  is  indebted  to  A,  $475,50,  to 
B,  8362,121,  to  C,  $250,87i,  and  to  D,  $140.  He  is  worth 
only  §614,25  :  to  how  much  is  each  entUled  ? 

12.  Brown,  Smith  &  Co.,  produce  dealers,  were  obliged  to 
make  an  assignment  on  account  of  a  falling  off  in  prices.  Their 
eftects  were  valued  at  $2544,  with  which  they  can  pay  but 
twenty  cents  on  the  dollar  :  how  much  did  they  owe  ? 

13.  Four  persons,  A,  B,  C,  and  D  agreed  to  do  a  piece  of 
work  for  $270.  Tiiey  \vere  to  do  the  work  in  the  proportions 
of  |,  A,  i,  and  yV  :  what  should  each  receive  for  his  work  ? 

14.  Three  persons  buy  a  piece  of  land  for  $4569,  and  the 
parts  for  which  they  pay  bear  tiie  following  proportions  to  each 
other,  viz.  :  the  sum  of  the  first  and  second,  the  sum  of  the  first 
and  third,  and  the  sum  of  the  second  and  third,  are  to  each 
other  as  i,  f  and  -^ :  how  much  did  each  pay,  and  what  part 
did  each  own  ? 

■      COMPOUND  PARTNERSHIP. 
When  the  Cause  of  Projit  or  Lass  is  Comj/ound. 
213.  When   the  partners  employ  their  capital   for  different 
periods  of  time,  the  profit  of  each  will  depend  on  two  circum- 
stances ;  first,  on  the  amount  of  capiUd  he  'puts  in  ;  and  secondly, 
on  the  time  it  is  continued  in  business. 


213  When  the  partners  employ  their  capital  for  different  periods  of 
time,  on  what  will  the  profit  depend  !  Will  the  cause,  then,  be  simple  or 
compound  1  To  what  will  it  be  equal  1  Give  the  rule  for  finding  each 
shard 


212  COMPOUND    PARTNERSHIP. 

Hence,  the  cause  of  the  loss  or  gain  will  be  compound,  and 
the  product  of  the  elements,  in  each  particular  case,  will  be  the 
cause  of  each  man's  gain  or  loss  ;  and  their  sum  AviU  be  the 
cause  of  the  entire  gain  or  loss. 

1.  A  put  in  trade  $000  for  4  months,  and  B,  $600  for  5 
months.     They  gained  $240  :  what  was  the  share  of  each  ? 

OPERATION. 

A,  $500  X  4  =  2000 

B,  GOO  X  5  :=  3000         ^^^^  ^    ^„„   ., 

-— —  .     <  2000    :  :    240    •    -f    ^^^  ^^• 

Hence,  the  following 

Rule. — MuUipIy  each  man^s  stock  bij  the  time  he  continued  it 
in  trade:  then  say,  as  the  sum  of  the  products  is  to  each  product, 
so  is  the  whole  gain  or  loss  to  each  man^s  share  of  the  gain  or 
loss. 

EXAMPLES. 

1.  Three  men  hire  a  pasture  for  $70,20  :  A  put  in  7  horses 
for  3  months  ;  B,  9  horses  for  5  months ;  and  C,  4  horses  for 
C  months  :  what  part  of  tlie  rent  should  each  pay  ? 

2.  A  commenced  business,  with  a  capital  of  $10000.  Four 
months  afterwards  B  entered  into  partnership  with  him,  and 
put  in  1500  barrels  of  tiour.  At  the  close  of  the  year  their 
profits  were  $5100,  of  which  B  was  entitled  to  $2100:  what 
was  the  value  of  tlie  flour  per  barrel  ? 

3.  On  the  1st  of  January,  1856,  A  commenced  business  with 
a  capital  of  $23000  ;  two  months  afterwards  ho  drew  out 
$1800;  on  the  1st  of  April  B,  entered  into  partnei-sliip  with 
liim,  and  put  in  $13500;  four  montiis  afterwards  hi'  drew  out 
$10000  ;  at  tlie  end  of  the  3'ear  their  profits  were  $8400  :  how 
much  ought  each  to  receive  ? 

■I.  'I'hrcc  i)er.-ons  received  intei'cst  to  the  amount  of  $798. 
A  put  out  $4000  for  12  months,  B,  $3000  for  15  months,  and 
C,  $5000  for  8  months  :  to  how  nuich  interest  was  each 
^Mititled  ? 


EXAMPLES. 


213 


5.  C,  D,  and  E,  form  a  copartneisliip ;  C's  sioek  is  in  trade 
3  montlis,  and  he  claims  yV  of  the  gain  ;  D's  stock  is  in  9 
months  ;  and  E  put  in  $7oG  for  4  months,  and  chiinis  l  of  the 
profits  :  how  much  did  C  and  D  put  in  ? 

6.  A  ship's  company  took  a  prize  worth  $20700,  which  was 
divided  among  them  according  to  their  pay  and  the  time  they 
had  been  on  board.  There  were  4  officers  receiving  $40  a 
month  each,  and  12  midsliipmen  receiving  $30  a  month  each, 
all  of  whom  had  been  on  board  6  months  ;  there  were  also  110 
sailoi's  receiving  $22  a  month  each,  and  who  had  been  on  board 
5  months  :  what  was  the  share  of  each  ? 

7.  Two  persons  form  a  partnership  for  one  year  and  six 
months.  A,  at  first,  put  in  $3000  for  9  months,  and  then  $1000 
more.  B,  at  first,  put  in  $4000,  and  at  the  end  of  the  first  year 
^500  more,  but  at  the  end  of  15  months  he  drew  out  $2000. 
At  the  end  of  12  months,  C  was  admitted  as  a  partner  witli 
$7333i.  The  gain  was  $7400  :  how  much  shouldeach  receive? 

8.  Four  persons  together  agreed  to  build  a  barn  for  $34G,50. 
A  worked  14  days,  12  hours  each  day;  B,  18  days,  10  hours 
each  day;  C,  15  days,  11  hours  each  day;  and  D,  20  days,  9 
hours  each  day :  how  much  should  each  man  receive  ? 

9.  In  a  certain  school,  premiums  to  the  value  of  $27  are  to 
be  distributed  in  tlie  following  manner :  The  premiums  are 
divided  into  three  grades.  The  value  of  a  premium  of  tlie 
first  grade  is  twice  the  value  of  one  of  the  second  ;  and  the 
value  of  one  of  the  second  grade  twice  that  of  the  third.  There 
are  6  to  receive  premiums  of  the  first  grade,  12  of  the  second, 
and  6  of  the  third :  what  will  be  the  value  of  a  single  premium 
of  each  grade  ? 

10.  Three  men  take  an  interest  in  a  mining  company.  A 
put  in  $480  for  6  months,  B,  a  sum  not  named,  for  12  months, 
and  C,  $320  for  a  time  not  named  :  when  the  accounts  Avere 
settled,  A  received  $G00  for  his  stock  and  profits,  B,  $1200  for 
his,  and  C,  $520  for  his  :  what  was  B's  stock,  and  C's  time  ? 


"214:  ^      PERCENTAGE. 


PERCENTAGE. 

214.  Pkrcentage  is  an  allowance  made  by  the  hundred,  and 
is  always  a  part  of  the  number  on  which  the  allowance  is  made. 

The  Base  of  percentage,  is  the  number  on  which  the  per- 
centage is  reckoned. 

215.  Percent  means  by  the  hundred:  thus,  1  per  cent 
means  1  for  every  hundred  ;  2  per  cent,  2  for  every  hundred ; 
3  per  cent,  3  for  every  hundred,  &c.  The  numbprs  denoting 
the  allowances,  1  per  cent,  2  per  cent,  3  per  cent,  &e.,  are  called 
rates,  and  may  be  expressed  decimally,  as  in  the  following 

TABLE. 


1  per  cent  is 

.01 

7  per  cent  is 

.07 

3  per  cent  is 

.03 

8  per  cent  is 

.08 

4  per  cent  is 

.04 

15  per  cent  is 

.15 

5  per  cent  is 

.05 

68  per  cent  is 

.68 

6  per  cent  is 

.06 

99  per  cent  is 

.99 

ALSO, 

100  per  cent  is  1 ;  for,  \^^  is  equal  to  1. 
150  per  cent  is  1.50 ;  for,  |^§  is  equal  to  1.50. 
140  per  cent  is  1.40 ;  for,  }^g  jg  equal  to  1.40. 
200  per  cent  is  2  ;  for,  ^%  is  equal  to  2. 

i  per  cent  is  .005 ;  for,  yoo  -t-  2  is  equal  to  .005. 
31  per  cent  is  .035;  for,  3i^l00  =  .03  +  .005  =  .035. 
5|  per  cent  is  .0575;  for,  5f^l00  =  .05  +  .0075  =  .0575. 
&c.  &c.  &c.  &c. 

examvles. 

1.  Write  decimally,  91  per  cent,  and  8^  per  cent. 

2.  Write  decimally,  12-i-  per  cent,  and  9|-  per  cent. 

3.  Write  decimally,  208  per  cent,  375  per  cent,  and  95  per  cent ' 

4.  Write  decimall}-,  6Gj  per  cent. 


S14.  What  is  percentage  1     What  is  the  base  1 

2 If).  What  docs  per  cent  mean  1     What  do  you  understand  by  3  pei 
aentl     Wliat  is  the  rate,  or  rato  per  cent  ? 


EXAilPLES. 


215 


216.    To  find  the  -percentage  of  any  number. 
1.  What  is  the  percentage  of  $450,  the  rate  being  6  per 
cent  ? 


OPERATION. 

450 
.06 


$27.00  Ans. 


Analysis. — The  rate  being  6  per  cent,  is 
expressed  decimally  by  .06.  We  are  then  to 
take  .06  of  the  base,  $450;  this  we  do  by 
multiplying  $450  by  .06,  giving  $27. 

Hence,  to  find  the  percentage  of  a  number, 

Multiply  the  nuinber  by  the  rate  expressed  decimally,  and  the 
product  tvill  be  the  percentage. 

EXAMPLES. 

1.  "What  is  the  percentage  of  $564,  the  rate  being  51  per 
cent? 


Note. — When  the  rate  cannot  be  re- 
duced to  an  exact  decimal,  it  is  most 
convenient  to  multiply  by  the  fraction, 
and  then  by  that  part  of  the  rate  which 
is  expressed  in  exact  decimals. 


OPERATION. 

564 

188  =  |-  per  cent. 
2820  =  5  per  cent. 


$3008  =  5-J-  per  cent. 
Find  the  percentage  of  the  following  numbers  : 


2.  i  per  cent  of  $1256. 

3.  -i  per  cent  of  §956,50. 

4.  -I  per  cent  of  475  yards. 

5.  |-  per  cent  of  o24:.6cwt. 

6.  A  pel-  cent  of  125.25/6. 

7.  If  per  cent  of  loQhush. 

8.  41  per  cent  of  $2000. 

9.  9  per  cent  of  186  miles. 

10.  10|-  per  cent  of  460  sheep. 

11.  5y*g-  per  cent  of  540  tons. 


12.  8|  per  cent  of  $3465,75. 

13.  12^  per  cent  of  126  cows. 

14.  50  per  cent  of  320  bales. 

15.  371  per  cent  of  127oyds. 

16.  95  per  cent  of  $4573. 

17.  105  per  cent  of  25005a?-. 

18.  11 2J-  per  cent  of  $4573. 

19.  250  per  cent  of  $5000. 

20.  305  per  cent  of  $1267,871. 

21.  500  per  cent  of  $3000. 


22.  What  is  the  difference  between  4f  per  cent  of  $1000 
and  71  per  cent  of  $1500  ? 


216.   How  do  3^ou  find  the  percentage  of  any  nunibrr  1 


216  PEKCENTAGE. 

23.  If  I  buy  895  gallons  of  molasses,  and  lose  17  per  cent 
by  leakage,  liow  much  have  I  left  ? 

24.  A  grocer  purchased  250  boxes  of  oranges,  and  found  that 
he  had  lost  in  bad  ones  18  per  cent :  how  many  full  boxes  of 
good  ones  had  he  left  ? 

25.  A  capitalist  wishes  to  invest  $25000  ;  he  invests  20  per 
cent  in  bank  stock,  371  per  cent  in  railroad  stock,  and  the 
remainder  in  bonds  and  mortgages  :  what  per  cent,  and  what 
amount  did  he  invest  in  the  latter  ? 

26.  A  man  bought  a  house  and  lot  for  $3250  ;  in  three  years 
time  it  increased  in  value  87^  per  cent :  what  was  its  value 
then  ? 

27.  A  farmer  having  $1572,75,  purchased  cows  with  25  per 
cent  of  it,  sheep  with  12^  per  cent  of  it,  and  lent  50  per  cent 
of  it  to  a  friend  :  how  much  had  he  left  ? 

217.    To  find  the  2Jer  cent  which  one  number  is  of  another. 

1.  What  per  cent  of  64  is  16  ? 

Analysis. — In  tliis  example  16  is  the  per-  operation. 

ccntage,  G4  is  the  base,  and  we  wish  to  find         16        1 
the  rate.     Since  the  percentage  is  equal  to         64       4       ' 
the  base  multiplied  by  the  rate  (Art.  216), 

the  rate   is   equal    to  the  percentage   divided  by  the  base ;    hence, 

1  fi       1 

-—  =  -  =  .25  ;  therefore,  the  rate  is  25  per  cent :  hence,  to  find  what 
64      4  '  '  ' 

per  cent  one  number  is  of  another, 

Divide  the  number  dinoting  the  percetitage  by  the  base,  and 
the  two  first  decimal  places  will  express  the  rate  per  cent. 
Notes. — 1.  The  base  is  generally  preceded  by  llic  word  of. 

2.  There  are  three  parts  in  i^ercentage :  1st.  The  base;  2d.  The 
rate;  and  3d,  their  product,  which  is  the  percentage. 

3.  The  percentage  divided  by  the  rate,  gives  the  base  ;  the  per- 
centage divided  by  the  6a.se,  gives  the  rate. 

217    How  do  you  find  the  per  cent  which  one  number  is  of  auotbor' 


EXAMPLES. 


EXAMPLES. 


1.  TVlmt  per  cent  of  10  dollars  is  2  dollars  ? 

2.  What  per  cent  of  32  dollars  is  4  dollars  ? 

3.  What  per  cent  of  40  pounds  is  3  pounds  ? 

4.  Seventeen  bushels  is  what  per  cent  of  125  bushels? 

5.  Thirty-six  tons  is  what  per  cent  of  144  tons  ? 
G.  What  per  cent  is  $84  of  $96  ? 

7.  What  per  cent  is  275  of  440  ? 

8.  What  per  cent  is  3  miles  of  400  miles  ? 

9.  Eleven  is  v/hat  per  cent  of  800  ? 

10.  One  hundred  and  four  sheep  is  Avhat  per  cent  of  a  drove 
of  312  sheep  ? 

11.  A  grocer  has  $325,  and  purchases  sugars  to  the  amount 
of  $121,87^  :  what  per  cent  of  his  money  does  he  expend  ? 

12.  Out  of  a  bin  containing  450  bushels  of  oats,  56^  bushels 
were  sold  :  what  per  cent  is   this    of  the  whole  ? 

13.  A  merchant  goes  to  New  York  with  $2500  ;  he  first  lays 
out  20  per  cent  for _  groceries,  and  then  expends  $1875  for  dry 
goods  :  what  per  cent  of  his  money  has  he  left  ? 

14.  Two  persons  invested  in  stocks  $4500  each ;  one  lost 
$562,50,  and  the  other  lost  $405  :  what  per  cent  more  on  the 
amount  invested,  did  one  lose  than  the  other  ? 

15.  A  and  B  engage  in  different  kinds  of  business  with 
$5400  capital  each  ;  A  gains  $1350,  and  B  loses  $540  the  first 
year  :  what  per  cent  is  B's  money  of  A's  ? 

218.  To  find  tlie  base  u'hen  Ihe  2>s^'centage  is  added  to  or  sub- 
traded  from  the  base. 

1.  Mr.  .Jones  buys  8  hogsheads  of  sugar,  sells  them  at  an 
advance  of  15  per  cent,  and  receives  $470  :  what  did  he  pay 
for  the  sugar  ? 


218.  How  do  you  find  the  base  when  the  percentage  is  subtracted  from 
tljc  base  1 


218  PERCENTAGE. 

Analysis. — The  amount  received,  $470  opekation. 

arises  from  adding  the  percentage  to  the         1.15)  470  ($400 
base;  that  if>j  it  arises  from  multiplying  the  470 

base  hy  1  +  pbu;  the  rate  per  cent ;  hence, 
to  find  the  base,  in  such  cases, 

Divide  (he  given  number  by  1  j>Zi<.?  the  roAe per  cent,  expressed 
decimally. 

2.  A  cask  of  Avine,  out  of  wliich  37  per  cent  had  leaked, 
was  found  to  contain  33.39  gallons  :  liow  many  gallons  did  the 
cask  contain  ? 

Analysis. — Thirty-seven  per  operation. 

cent  denotes  .37  of  the  capacity  1  —.37  per  cent  =  .63  per  cent, 
of  the  cask ;  and  hence,  the  part 

of  the  cask  that  is  filled  is  de-  .63)   33.39   (53  gallons, 
noted   by   1  -  .37  =  .63.      But  31  5 
.63  of  the  cask  contains  33.39  1  89 
gallons  :    therefore,    the   entire  1  89 
cask  will  contain  as  many  gal- 
lons as  .63  is  contained  times  in  33.39.  viz.,  53  ;  hence,  to  find  the 
base,  in  such  cases, 

Divide  the  given  number  by  the  difference  between  1  und  the 
rate  p)£r  cent,  expressed  decimally. 

EXA3IPI.es. 

1.  A  farmer  bought  40  sheep,  and  after  keeping  them  for 
one  year,  sold  them  at  an  advance  of  55  per  cent,  and  received 
$248  :  what  did  he  pay  for  tlie  sheep  per  head  ? 

2.  A  merchant  bought  a  lot.  of  goods  and  marked  ihcni  at  .in 
advance  of  20  per  cent :  -vvhen  sold,  he  found  that  they  brought 
liini  $6835,50  :  what  did  the  goods  cost  him  ? 

3.  A  son,  who  inherited  a  fortune,  spent  371  per  oent  of  it, 
when  lie  found  tliat  lie  had  only  $31250  remaining:  what  was 
the  amount  of  his  fortune  ? 

4.  A  grocer  purchased  a  lot  of  teas  and  sugar,  on  wiiicli  he 
lost  16  i)(!r  cent,  by  selling  them  for  .^4200  :  wliat  did  he  pay 
fni-  the  goods  ? 


INTEREST.  219 

INTEREST. 

219.  Interest  is  a  payment  for  the  use  of  money. 
Principal  or  Base,  is  the  money  on  which  interest  is  paid. 

Amount  is  the  sum  of  the  Principal  and  Interest. 

For  example  :  If  I  borrow  $100  for  1  year,  and  pay  7  dol 
lars  for  the  use  of  it ;  then, 

100  dollars  is  the  Principal  or  Base, 

7  dollars  is  the  Interest,  and 
107  dollars  is  the  Amount. 

Rate  is  the  ratio  of  the  principal  to  the  interest,  when  the 
time  is  1  year.  Thus,  .07  is  tlie  rate  in  the  example,  being 
the  ratio  of  $100  to  §7.  This  rate  is  read,  7  -per  cent ;  that 
is,  $7  for  every  hundred  :  tlie  term  per  cent,  means  by  the 
hundred,  and  tlie  term,  per  annum,  by  the  year. 

Hence,  there  are  four  simple  parts  :  1st.  Principal ;  2nd. 
Rate  ;  ord.  Time ;  4th.  Interest. 

CASE   I. 

220.  To  find  the  interest  of  any  principal  for  one  or  more 
years. 

1.  What  is  the  interest  of  $3920  for  2  years,  at  7  per  cent  ? 

Analysis. — The  rate  of  interest  being                operation. 
7  per  cent,  is  expressed  decimally  by  .07  :               S3920 
hence,  each  dollar,  in  1    year,  ayIII   pro-                   .07  rate. 
duce  .07  of  itself,  and   S3920  will  pro-          $274,40  int.  for  1  year, 
duce  .07  of  S3920,  or  $274,40.     There-                   '    3  No.  of  years, 
fore,  $274,40  is   the  interest  for  1  year,           $548,80  interest, 
and  this   interest  multiplied  by  2,  gives 
the  interest  for  2  years  :  hence,  the  following 
— -^ _ « ■ 

219.  What  is  interest  1  What  is  principalT  What  is  amount  1  What 
is  Tate  of  interest  1     What  does  per  annum  mean  1 

220.  How  do  you  find  the  interest  of  any  principal  for  any  number  of 
years  1     Give  the  analysis. 


220  INTEREST. 

RtjLE. — Multiphj  the  principal  by  the  rate,  expressed  decimaUy 
and  the  product  by  the  number  of  years. 

EXAMPLES. 

1.  Wliat  is  the  interest  of  $675  for  1  year,  at  6i  per  cent? 

OPERATION. 

$675 
Analysis. — ^We  first  find  the  interest  Qgi 

at  \  per  cent,  and  then  the  interest  at  ^Z" o  ^  per  cent 

6  per  cent;  the  sum  is  the  interest  at  4050     6  per  cent. 

6^  per  cent.  $43,875  fil  per  cent. 

2.  AYhat  is  the  interest  of  $871,25,  for  1  year,  at  7  per  cent  ? 

3.  What  is  the  interest  of  $535,50,  for  7  years,  at  6  per 
cent  ? 

4.  What  is  the  interest  of  $1125,885,  for  4  years,  at  8  per 

cent  ? 

5.  What  is  the  interest  of  -§789, 74,  for  12  years,  at  5  per  cent  ? 

6.  What  is  the  intex-est  of  6^2500,  for  7  years,  at  7i  per  cent  ? 

7.  What  is  the  interest  of  83153,82,  for  2  years,  at  A\  per 

cent? 

8.  What  is  the  amount  of  $199,48,  for  16  years,  at  7  per 

cent  ? 

9.  What  is  the  amount  of  $897,50.  for  3  years,  at  8  per  cent  ? 

10.  What  is  the  interest  of  $982,35,  for  4  years,  at  63-  per 
cent? 

11.  What  is  the  amount  of  ^1500,  for  5  years,  at  5\  per  cent  ? 

12.  What  is  the  interest  of  $1914,10,  for  6  years,  at  3|-  per 
cent  ? 

13.  What  is  the  interest  of  $350,  for  21  years,  at  10  per 
cent  ? 

14.  What  is  the  amount  of  628,50,  for  5  years,  at  121  per 
cent  ? 

15.  What  is  the  amount  of  §75,50,  for  10  years,  at  6  per 
cent? 

16.  What  is  the  amount  of  85040,  for  2  years,  at  1\  per 
cent  ? 


INTEREST.  221 

Note. — When  taerc  are  years  and  months,  and  the  months  are 
an  aliquot  part  of  a  year,  multiply  the  interest  for  1  year  by  the  years 
and  the  months  reduced  to  the  fraction  of  a  year. 

EXAMPLES. 

1.  What  is  the  interest  of  $119,48  for  2  years  6  months,  at  7 
per  cent  ? 

2.  What  is  the  interest  of  $250,60  for  1  year  9  months,  at  6 
per  cent  ? 

3.  Yfhat  is  the  interest  of  $956  for  5  years  4  months,  at  9 
per  cent  ? 

4.  Wliat  is  the  amount  of  $1575,20  for  3  years  8  months,  at 
7  ])CY  cent  ? 

5.  What  is  the  amount  of  $5000  for  2  years  3  months,  at  5^ 
per  cent  ? 

6.  What  is  the  interest  of  $1508,20  for  4  years  2  months,  at 
10  per  cent  ? 

7.  What  is  the  interest  of  $75  for  6  years  10  months  at  12^- 
per  cent  ? 

8.  Wliat  is  the  amount  of  $125  for  5  years  6  months,  at  4|- 
per  cent  ? 

CASE   II. 

221.  To  find  the  interest  on  a  given  principal  for  any  rate 
and  time. 

1.  What  is  the  interest  of  $1752,96  at  6  per  cent,  for  2  years 
4  months  and  29  days  ? 

Analysis. — The  interest  for  1  year  is  the  product  of  the  principal 
multiplied  by  the  rate.  If  the  interest  for  1  year  be  divided  by  12, 
the  quotient  will  be  the  interest  for  1  month :  if  the  interest  for  1 
month  be  divided  by  30,  the  quotient  will  be  the  interest  for  1  day. 

The  interest  for  2  years  is  2  times  the  interest  for  1  year  :  the 
interest  for  4  monlht-,  4  times  the  interest  for  1  month ;  and  the 
nitere.st  for  29  days,  29  times  the  interest  for  1  day. 


221.  How  do  you  find  the  interest  for  any  time  and  rate  "i     How  do  you 
fiud  the  interest  for  years,  months,  and  days  by  the  second  method  T 


222  PEKOENTAGK. 

$1752;96  OPERATION. 

.06 


12)105,1776  int.  for  1  2/r.     Sl05,l776     X2    =$210,3552    2yr. 
30)8.7  (i48  int.  for  Imo.  8,7648    X4    =      35,0592    4?no. 

,29216  int.  for  Ida.  0,29216X29=        8.47264  29c/a 

Total  interest,     ©253,88704 
Hence,  we  have  the  following 
EuLE. — I.  Find  the  interest  for  1  year  : 

II.  Divide  t/iis  interest  by  12,  and  the  quotient  will  be  the 
interest  for  1  month  : 

III.  Divide  the  interest  for  1  month  by  30,  and  the  quotient 
will  be  the  interest  for  1  day. 

IV.  Multiply  the  interest  for  1  year  by  the  number  of  years, 
the  interest  for  1  month  by  the  number  of  months,  and  the  inte- 
rest  far  1  day  by  the  number  of  days,  and  the  sum  of  the  products 
will  be  the  required  interest. 

Note. — This  method  of  computing  interest  for  day.s,  is  the  one  in 
general  use.  It  supposes  tlic  month  to  contain  30  days,  or  the  year 
360  days;  whereas,  it  actually  contains  365  days. 

To  find  the  exact  interest  for  1  day,  we  must  regard  the  month  as 
containing  3j;_5  days  =  30-5-  days;  and  this  is  the  number  by  which 
the  interest  for  one  month  should  be  divided,  in  order  to  find  the  exact 
interest  for  one  day.  As  the  divisor,  commonly  used,  is  too  small,  the 
interest  found  for  1  day,  is  a  trifle  too  large.  If  entire  accurac--  s 
required,  the  interest  for  the  days  must  be  diminished  by  its  -g-^-j 
part  =  Tf  3-  part. 

2d  method. 

222.  There  is  another  rule  resulting  from  the  last  analysis  which 
is  regarded  as  the  best  general  method  of  computing  intei'cst. 

Rule. — I.  Find  the  interest  for  1  year  and  divide  it  by  12: 
the  quotient  will  be  the  interest  for  1  month. 

II.  Multijily  the  interest  for  1  month  by  the  time  expressed  in 
months  and  decimal  parts  of  a  month,  and  the  product  will  be 
the  required  interest. 

Note. — Since  a  month  is  reckoned  at  30  days,  any  number  of  days 
is  rulutcd  to  decimals  of  a  month  by  dividing  the  number  uf  days  by  3. 


INTERESl  223 


EXAMPLES. 

1.   What  is  the  interest  of  $655,  for  3  years  7  months  and 

13  days,  at  7  per  cent  ? 

OPERATION. 

Syrs.  =  SGmos. 

$655 

7vios. 

.07 

13da.   =      A^mos. 

12)45.85              int.  for  1  year. 

Time  —  43  Aminos. 

3.82083  +   int.  for  1  month. 

43.4^     time  in  months. 

127361 

1528332 

1146249 

1528332 

165,951383  Ans. 

2.  What  is  the  interest  of  $358,50,  for  1  year  8  months  and 
6  days,  at  7  per  cent  ? 

3.  What  is  tlie  interest  of  $1461,75,  for  4  years  9  months 
and  15  days,  at  6  per  cent  ? 

4.  What  is  the  interest  of  $1200,  for  2  years  4  months  and 
12  days,  at  7i  per  cent  ? 

5.  What  is  the  interest  of  $4500,  for  9  months  and  20  days, 
at  5  per  cent  ? 

6.  What  is  the  interest  of  $156,25,  for  10  months  and  18 
days,  at  8  per  cent  ? 

7.  What  is  the  interest  of  6640,  for  3  years  2  months  and 
9  days,  at  6^  per  cent  ? 

8.  What  is  the  interest  of  $276,50,  for  11  months  and  21 
days,  at  10  per  cent  ? 

9.  What  is  the  amount  of  $378,42,  for  1  year  5  months  and 
3  days,  at  7  per  cent  ? 

10.  What  is  the  amount  of  $1250,  for  7  months  and  21  days, 
a.t  101^  per  cent  ? 

11.  What  is  the  interest  of  $6500,  for  2  months  and  10  days, 
at  9^  per  cent  ? 

12.  What   is   the  interest  of  $70,50,  for   10  years   and  10 
months,  at  5^  per  cent  ? 


22-i  PEEOENTAGE. 

13.  What  is  the  amount  of  $45,  for  12  years  and  27  days, 
at  6j  per  cent  ? 

14.  What  will  $100  amount  to  in  15  years  and  6  months,  if 
put  at  interest  at  4  jier  cent  ? 

15.  How  much  will  $475,50  gain  in  5  years  9  months  and 
24  days,  at  8  per  cent  ? 

IG.  What  will  be  the  interest  of  $45G0,  for  14  months  and 
19  days,  at  7  per  cent  ? 

17.  Wha^  will  $128,371  amount  to  in  10  months  and  27 
days,  at  G  per  cent  ? 

18.  What  is  the  interest  of  $264,52,  for  2  years  8  months 
and  14  days,  at  6  per  cent  ? 

19.  What  is  the  amount  of  $76,50,  for  1  year  9  months  and 
12  days,  at  6  per  cent? 

20.  What  Avill  be  the  interest  for  3  years  3  months  and  15 
days,  of  $241,60,  at  7  per  cent? 

21.  What  is  the  interest  of  $5600,  for  30  days,  at  7  per  cent? 

22.  What  will  $8450  amount  to  in  GO  days,  at  10  per  cent  ? 

23.  What  is  the  interest  of  $4000,  for  1  month  and  6  days, 
at  9  per  cent  ? 

24.  What  will  be  the  amount  of  $87,60,  from  Sept.  9th, 
1852,  to  Oct.  10th,  1853,  at  6i  per  cent? 

25.  What  will  be  due  on  a  note  of  $126,75,  given  July  8t;h, 
1854,  and  payable  April  25th,  1858,  at  7  per  cent  ? 

26.  What  is  the  interest  of  ^350,  from  Jan.  1st,  1856,  to  15tli 
of  Sept.  next  following,  at  5-]-  per  cent  ? 

27.  Gave  a  note  of  $560,40,  March  14th,  1855,  on  interest, 
after  90  days  :  what  interest  was  due  Dec.  1st,  1856,  at  10  per 
cent  ? 

28.  Find  the  interest  of  $1256,  for  11  months  and  9  days, 
at  6  per  cent. 

29.  What  is  the  amount  of  $745,40  at  5  per  cent  interest, 
being  reckoned  from  the  5lh  day  of  the  10th  month  of  1850, 
to  the  10th  day  of  tlie  5th  month  of  1854  ? 

30.  Sept.  10th,  James  Trusty  borrowed  of  Peter  Credit 
$250,  and  IMarch  4th,  1853,  he  borrowed  $500  more,  agreeing 


rNTEKEST.  225 

to  pay  7  per  cent  interest  on  the  whole ;  what  was  the  amount 
of  his  indebtedness  Jan.  1st,  1854  ? 

31.  Ordered  dry  goods  of  A.  T.  Stewart  &  Co.,  at  different 
times,  to  the  following  amounts,  viz.,  Jan.  1st,  1854,  $254 ; 
March  15th,  1854,  $154,G0  ;  April  20th,  1854,  $424,25  ;  and 
June  3d,  1854,  $75,50.  I  bought  on  time  at  6  per  cent  in- 
terest :  what  was  the  whole  amount  of  my  indebtedness  on  the 
tirst  day  of  Sept.  following  ? 

32.  If  I  borrow  $475,75  of  a  friend  at  7  per  cent,  what  will 
I  owe  him  at  the  end  of  8  months  and  a  half? 

33.  In  settling  with  a  merchant,  I  gave  my  note  for  $127,28, 
due  in  1  year  9  months,  at  6  per  cent :  what  must  be  paid  when 
the  note  falls  due  ? 

34.  A  person  buying  a  piece  of  property  for  $4500,  agreed 
to  pay  for  it  in  thi'ee  equal  annual  instalments,  with  interest  at 
6-^  per.  cent :  what  was  the  entire  amount  of  money  he  paid  ? 

35.  A  mechanic  hired  a  journeyman  for  9  months  at  $40  a 
mouth,  to  be  paid  monthly  ;  at  the  end  of  the  time  he  had 
paid  nothing;  he  then  settled,  allowed  interest  at  7  per  cent, 
and  gave  his  note,  on  interest,  due  in  1.  year  4  months  and  15 
days  :  what  will  he  pay  when  his  note  falls  due  ? 

36.  A  person  owning  a  part  of  a  woollen  factory,  sold  his 
share  for  $9000.  The  terms  were,  one-third  cash,  on  delivery 
of  the  property,  one-half  of  the  remainder  in  6  months,  and  the 
rest  in  12  months,  M'ith  7^  per  cent  interest:  what  was  the 
whole  amount  paid  ? 

NOTES. 


$382,50  Chicago,  January,  1st,  1856. 

1.  For  value  received  I  promise  to  pay  on  the  10th  day  of 
June  next,  to  C.  Hanford  or  order,  the  sum  of  three  hundred 
and  eighty-two  dollars  and  fifty  cents,  with  interest  fi'om  date, 
at  7  per  cent. 

$612  Baltimore,  January  1st,  1856. 

2.  For  value  received  I  promise  to  pay  on  the  4th  of  July, 
1858,  to  Wm.  Johnson  or  order,  six  hundred  and  twelve  dollars 
with  interest  at  6  per  cent  from  the  1st  of  March,  1856. 

John  Liberal. 


226  PERCENTAGE. 


$3120  Charleston,  July  3d,  1855. 

3.  Six  months  aftei*  date,  I  promise  to  pay  to  C.  Jones  or 
order,  three  thousand  one  hundred  and  twenty  dollars  with 
interest  from  the  1st  of  January  last,  at  7  per  cent. 

Joseph  Sprinps. 


$786,50  New  York,  July  7th,  1851. 

4.  Twelve  months  after  date,  I  promise  to  pay  to  Smith  & 
Baker  or  order,  seven  hundi-ed  and  eighty-six  -j^^  dollars  for 
value  received  with  interest  from  December  3d,  1851,  at  8  per 
cent.  ""  Silas  Day. 


$4560,72  Cincinnati,  March  10th,  1856. 

5.  Nine  months  after  date,  for  value  received,  I  promise  to 
pay  to  Redfield,  Wright  &  Co.  or  order,  four  thousand  five; 
hundred  and  sixty  -J-^^  dollars  with  interest  after  6  monthi?,  at 
7  per  cent.  Frederick  Stillman. 


$1854,83  Boston,  July  17th,  1856. 

6.  Eleven  months  after  dale,  for  value  received,  we  promise 
to  pay  to  the  order  of  Fondy,  Burnap  &  Co.,  one  thousand  eight 
hundred  and  fifty-four  y^q  dollars  with  interest  from  May  13th, 
1856,  at  6  per  cent.  Palmer  <C-  Blake. 

POUNDS,    SUILLINGS    AND    PENCE. 

223.  To  find  the  interest,  when  the  principal  is  pounds,  shil- 
lings and  pence. 

I.  Reduce  the  shillinr/s  and  2:>ence  to  the  decimal  of  a  pound 
(Art.  164). 

II.  Then  Jind  the  interest  as  though  the  sioii  were  dollars  and 
cents  ;  after  which  reduce  the  decimal  part  of  the  answer  to  shil 
linys  and  pence  (Art.  165). 


223.  How  do  you  find  tho  interest  when  the  principal  is  pounds,  shil- 
lings and  pence ' 


ESTTKUEST.  .  227 

EXAMPLES. 

1.  What  is  the  interest,  at  6  per  cent,  of  £27  15s.  9d.  for  2 

years  ? 

£27  15s.  9d.  z=  £27.7875. 

£27.7875  x  .06  x  2  =  £3.3345  interest. 

£3.3345  r=  £3  6s:  S\d.  Ans. 

2.  "What  is  the  interest  on  £203  18s.  Gd.,  at  6  per  cent,  for 
3  years  8  months  16  days  ? 

S.^What  is  the  interest  of  £215  13s.  8:/.,  at  6  per  cent,  for  3 
years  6  months  and  9  days  ? 

4.  Wliat  is  the  interest  of  £1543  10s.  6c/.,  for  2  years  and  a 
half,  at  4  per  cent  ? 

5.  What  is  the  amount  of  £1047  3s.,  for  \p\  4mo.  15da.,  at 
6  per  cent.  ? 

6.  What  is  the  interest  on  £511   Is.  4c/.,  at  6  per  cent  per 
annum,  for  6yr.  G))io.  ? 

7.  What  is  the  interest  on  £161   15s.  Sd.,  at  6  per  cent,  for 
Smo.  loda.  ? 

PROBLEMS  ]N  INTEREST. 

224,  In  every  question  of  interest  there  are  four  parts  :  1st. 
Principal ;  2d.  Rate ;  3d.  Time  ;  ^nd  4th.  Interest.  If  any 
three  of  these  parts  are  known,  the  4th  can  be  found. 

1.  At  what  rate  per  cent  must  $325  be  put  at  interest  for 
1  year  and  6  months,  to  produce  an  interest  of  $34,125  ? 

Analysis. — The  principal,  multiplied  by  opekation. 

the  rate  expressed  decimally,  multiplied  by      Principal, 
the  time   in  years,  is  equal  to  the  interest      Rate,  12 

(Art.  221)  ;  and  when  the  time  is  expressed      Time,  Interest, 

in  montlis  and  decimals  of  a  month,  the  same      in  months, 
product  is  equal  to  ]  2  times  the  interest  (Art. 221).  Hence, 

224.  How  many  parts  are  there  m  every  question  of  interest]  How 
many  of  these  must  be  known  before  the  remainder  can  be  found  !  How  do 
you  find  the  interest  when  you  know  the  Principal,  Rate,  and  Time  1  How 
do  you  find  the  Principal  when  you  know  the  interest,  rate,  and  time  1  How 
do  vo'i  apply  the  formula  to  any  ciuie  ' 


223  PEEOKNTAQE. 

I.  When  the  time  ts  in  months,  the  product  of  the  principal 
rate  and  time  will  be  equal  (o  12  times  the  interest. 

II.  When  tioo  of  these  parts  and  the  interest  are  given,  12  times 
the  interest  divided  hy  the  product  of  the  given  parts  will  he  equal 
to  the  other  part. 

Note  — Let  this  formula  be  ^Yritten  on  the  black  board,  or  slate, 
and  all  the  examples  worked  by  it. 

To  apply  the  rule  to  the  above  example,  operation. 


place  $325  for  the  principal,  x  for  the  rate,  $325 

18  (months)  for  the  time,  and   $34,125  for  ^. 

the  interest.     Cancelling  and   dividing,  we  ^^ 


2 

n 

$34,125 


find  a:  =  .07  :  or,  the  rate  is  7  per  cent.  o  -  i .-,-      c 

325x3 
Ans.  7  per  cent. 

EXAMPLES. 

1.  What  principal,  at  G  per  cent,  will  in   9  months   give  an 
interest  of  $178,9552  ? 

2.  The   interest   for  2  years  and  6  months,  at  7  per  cent,  is 
$76,9G5:  what  is  the  principal? 

3.  What  sum  must  be  invested,  at  6  per  cent,  for  10  months 
and  15  days,  to  produce  an  interest  of  §327,3249  ? 

4.  If  my  salary  is  §1500  a  year,  what  sum  invested  at  5  per 
cent,  will  pay  it  ? 

■  5..  What  sum  put  at  interest  for  4  years  and  3  months,  at 
7  per  cent,  will  gain  $283,3914? 

6.  The  interest  of  $2100  for  3  years  1  month  and  18  days  is 
$4G0,G0  :  what  is  the  rate  per  cent? 

7.  A  man  invests  $5420   in  Eailroad  stock,  and  receives  a 
semi-annual  dividend  of  $244,17  :  what  is  the  rate  per  cent? 

8.  A  person  owning  property  valued  at  $2470,80,  rents  it  for 
1  year  and  10  months  for  $452,98  :  what  per  cent  does  it  pay? 

9.  At  what  rate  per  cent  must  $3456  be  loaned  for  2  years 
7  months  and  24  days,  to  gain  $503,712  ? 

10.  If  I  build  a  hotel  at  a  cost  of  $5GO0O,  and  rent  it  for 
67000  a  year,  what  per  cent  do  I  receive  for  the  investment  ? 


INTKKEST.  229 

11.  The  interest  on  $1119,48,  at  7  per  cent,  is  $195,900 : 
what  is  the  time  ? 

12.  A  man  received  $47,25  for  the  use  of  11750  ;  the  rate 
of  interest  being  9  per  cent :  what  was  the  time  ? 

13.  How  long  will  it  take  ^7500  to  amount  to  $7850,  at 
3J  per  cent  per  annum  ? 

14.  How  long  will  it  take  $500  to  double  itself,  at  6  per  cent, 
simple  interest  ? 

15.  Wishing  to  commence  business,  a  friend  loaned  me  $3720, 
at  6^  per  cent,  which  I  kept  until  it  amounted  to  $5009,60  : 
how  long  did  I  retain  it  ? 

16.  I  borrowed  $700  of  my  neighbor  for  1  year  and  8  months, 
at  6  per  cent ;  at  the  end  of  the  time  he  borrowed  of  me  ^750 
how    long   must   lie    keep    it   to  cancel   the    amount  of  inte- 
rest I  owed  him? 

PARTIAL    PAYMENTS. 

225.  We  shall  now  give  the  rule  established  in  New  York 
(See  Johnson's  Chancery  Reports,  Vol.  I.,  page  17,)  for  com- 
puting the  interest  on  a  bond  or  note,  when  partial  payments 
have  been  made.  The  same  rule  is  also  adoiJted  in  Massachu- 
setts, and  in  most  of  the  other  states. 

Rule. — I,  Compiite  the  interest  on  the  principal  to  the  time 
of  tlic  Jirst  payment,  and  if  the  payment  exceed  this  interest,  add 
the  interest  to  the  principal,  and  from,  the  sum  subtract  the  pay- 
ment :  the  remainder  forms  a  new  principal. 

11.  But  if  the  payment  is  less  than  the  interest,  take  no  notice 
of  it  until  other  payments  are  7nade,  tvhich  in  all,  shall  exceed 
the  interest  computed  to  the  time  of  the  last  payment :  then  add 
the  interest,  so  computed,  to  the  principal^  and  from  the  sum 
subtract  the  sum  of  the  payments  :  the  remainder  will  form  a 
new  princip>al  on  which  interest  is  to  be  computed  as  before. 


225.  What  is  the  rule  for  partial  payments  1 


230  PERCENTAGE. 


EXAMPLES. 


$349,998  Richmond,  Va.,  May  1st,  1846 

1.  For  value  received  I  promise  to  pay  James  Wilson  or 
order,  three  hundred  and  forty-nine  dollars  ninety-nine  cents 
and  eight  mills  with  interest,  at  6  per  cent. 

James  Paywell 

On  this  note  were  endorsed  the  following  payments  : 

Dec.  25th,  1846,  received  $49,998 
July  10th,  1847,         "            4,998 
Sept.  Ist.,  1848,         "          15,008 
June  14th,  1849,         "          99,999 
What  was  due  April  15th,  1850  ? 
Principal  on  int.  from  May  1st,  1846,  -     -     -     -     $349,998 
Interest  to  Dec.  25th,  1846,  time  of  first  pay- 
ment, 7  months  24  days, 13,649  -f- 

Amount,     -     -     -     -     $363,647  •-{- 

Payment  Dec.  25th,  exceeding  interest  then  due  §  49,998 

Remainder  for  a  new  principal $313,649 

Interest^of  ^313,649   from   Dec.  25th,  1846,  to 

June  14th,  1849,  2  years  5  months  19  days        §46,472  + 

Amount,     -     -     -  §360,121 

Payment,  July  10th,  1847,  less~» 

than  interest  then  due,     -      )  ' 

Payment,  Sept.  1st,  1848,     -     -     -     15,008_ 

T^^^^^""^'     ". I    "i^20,006" 

less  than  interest  then  due    > 

Payment,  June  14lh,  1849,      -     -       99,999 

Their  sum  exceeds  the  interest  then  due  -  -  $120,005 
Remainder  for  a  new  principal,  June  14th, 

1849, 240,116 

Interest  of  $210,116  from  June  14lh,  1849, 

to  April  15th,  1850,  10  months  1  day,  $  12.015 

ToliU  due,  April  15th,  1850,     -     -     -  ¥252,T6r-f 


PARTIAL    PAYMENTS.  281 


$6478,84:  New  Haven,  Feb.  6th,  1850. 

2.  Foi'  value  received  I  promise  to  pay  William  Jenks  or 
order,  six  thousand  four  hundred  and  seventy-eight  dollars  and 
eighty-four  cents  with  interest  from  date,  at  6  per  cent. 

John  Stewart. 

On  this  note  were  endorsed  the  following  payments  : 

May  16th,  1853,  received  $545,76 
May  16th,  1855,         "  1276 

Feb.  1st,     1856,         "  2074,72. 

What  remained  due  August  11th,  1857  ? 

3.  A's  note  of  $7851,04  was  dated  Sept.  5th,  1851,  on  which 
were  endorsed  the  following  payments,  viz.  : — Nov.  loth,  1853, 
$416,98;  May  10th,  1854,  $152:  what  was  due  March  1st, 
1855,  the  interest  being  6  per  cent  ? 


$8'J74,56  New  York,  Jan.  3d,  1854. 

4.  For  value  received  I  promise  to  pay  to  James  Knowles 
or  order,  eight  thousand  nine  hundred  and  seventy-four  dollars 
and  fifty-six  cents,  with  interest  from  date  at  the  rate  of  7  per 
cent.  Stephen  Jones, 

On  this  note  are  endorsed  the  following  payments  : 

Feb.  16th,  1855,  received  $1875,40 
Sept.  15th,  1856,         "  3841,26 

Nov.  11th,  1857,         "  1809,10 

June    9th,  1858,         "  2421,04. 

What  will  be  due  July  1st,  1858  ? 


S345,50  Buffalo,  Nov.  1st,  1852. 

5.  For  value  received  I  promise  to  pay  C.  B.  Morse  or  order, 
three  hundred  and  forty-five  dollars  and  fifty  cents  with  interest 
from  date,  at  7  per  cent.  Ja^in  Dor. 

On  this  note  are  the  following  endorsements  : 


232 


PEKCENTAGB. 

Juno  20tli, 

1853, 

received 

$75 

Jan.   12th, 

1854, 

ii 

10 

March  3d, 

1855, 

a 

15,50 

Dec.  13th, 

1856, 

u 

52,75 

Oct.  14th, 

1857, 

a 

106,75 

"What  will  there  be  due  Feb.  4th,  1858  ? 


$450  Mobile,  Oct.  19th,  1850. 

6.  For  value  received  we  jointly  and  severally  promise  to 
pay  Jones,  Mead  &  Co.  or  order,  four  hundred  and  fifty  dollars 
on  demand  with  interest,  at  8  per  cent.  Manning  c5  Bros. 
The  following  endorsements  were  made  on  this  note  : 
Sept.  25,  1851,  received  $85,60  ;  July  10,  1852,  received 
S20  ;  June  6,  1853,  received  S150,45  ;  Dec.  28,  1854,  received 
§25,12J  ;  May  5,  1855,  received  $169  :  what  was  due  Oct. 
18,  1857  ? 

LEGAL  INTEREST. 

226.  Legal  Interest  is  the  interest  which  the  law  permits  a 
person  to  receive  for  money  which  he  loans,  and  the  laws  do 
not  favor  the  taking  of  a  higher  rate.  In  most  of  the  states 
the  rate  is  fixed  at  6  per  cent ;  in  New  York,  South  Carolina 
and  Georgia,  it  is  7  ;  and  in  some  of  the  states  the  rate  is  fixed 
as  high  as  10  per  cent. 

COMPOUND  INTEREST. 

227.  Compound  Interest  is  when  the  interest  on  a  principal, 
computed  to  a  given  time,  is  added  to  the  principal,  and  the 
interest  then  computed  on  this  amount,  as  on  a  new  principal. 

NoTK. — The  laws  do  not  favor  the  payment  of  compound  interest. 
Ill  most,  of  the  states,  an  agreement  to  pay  compound  inlorc.-st,  could 
uot  be  enforced. 

226.  What  i.s  legal  interest  1 

227.  What  is  compound  interest  T     How  do  you  computo  it ' 


EXAMPLES.  233 

Rule. — Compute  the  interest  to  the  time  at  which  it  becomes 
due  ;  then  add  it  to  the  j^rincijxd  and  compute  the  interest  on  the 
amount  as  on  a  new  j^^'incijKil :  add  the  interest  again  to  the 
jyrincipal  and  compute  the  interest  as  hefore  ;  do  the  same  for  all 
the  times  at  which  payments  of  interest  become  due  ;  from  the 
last  result  subtract  the  2^^'irici2}al,  and  the  remainder  will  be  the 
comp>ound  interest. 

EXAMPLES. 

1.  What  will  be  tlie  compound  interest  of  $3750  for  4  years, 
at  7  i^er  cent  ? 

$3750,000        principal  for  1st  year. 

$3750  X  .07  =     2G2,500       interest    for  1st  year. 

4012,500       principal  for  2tl      " 

64012,50  X  .07  =     280,875       interest    for  2d      « 

42y3,'375       principal  for  3d      " 

$4293,375  x  .07  =     300,536  +  interest    for  3d      " 

4593^911  +  principal  for  4th     '* 
$4593,911  x  .07  =     321,573  +  interest    for  4th     « 

4915,484  +  amount  at  4  years. 
1st  principal     3750,000 
Amount  of  intei-est      $1105,484  + 

2.  What  will  be  the  compound  interest  of  1175  for  2  years, 
at  7  per  cent  ? 

3.  What  will  be  the  amount  of  $240  at  compound  interest, 
for  4  years,  at  5  per  cent  ? 

4.  What  will  be  the  compound  interest  of  8300,  for  three 
years,  at  6  per  cent  ? 

5.  What  will  be  the  compound  interest  of  $590,74,  at  6  per 
cent,  for  2  years  ? 

6.  What  will  be  the  compound  interest  of  $500,  for  2  year, 
nt  8  per  cent? 

7.  What  will  be  the  compound  interest  of  $3758,50,  for  3 
years,  at  7  per  cent  ? 

8.  What  will  be  the  compound  interest  of  $95037,50,  for  7 

years,  at  0  per  cent  ? 

11 


234 


COMPOUND    INTEREST. 


Note. — The  operation  is  rendered  much  shorter  and  easier,  by 
taking  the  amount  of  1  dollar  for  any  time  and  rate  given  in  the 
following  table,  and  multiplying  it  by  the  given  principal ;  the  pro- 
duct will  be  the  required  amount,  from  which  subtract  the  given 
principal,  and  the  result  will  be  the  compound  interest. 

TABLE — Showing  the  amount  of  Si  or  £1,  compound  interest,  from  1 
year  to  20  years,  and  at  the  rate  of  3,  4,  5,  6,  and  7  per  cent. 


Years 

3  per  cent. 

4  per  cent. 

5  per  cent. 

0  per  cent. 

7  per  cent 

Years. 

1 

1.03000 

1.04000 

1.05000 

1.06000 

1.07000 

1 

O 

1.06090 

1.18160 

1.102.50 

1.12360 

1.14490 

2 

3 

1.09272 

1.12486 

1.15762 

1.19101 

1.22.504 

3 

4 

1.12.550 

1.16985 

1.215.50 

1.26247 

1.31079 

4 

5 

1.15927 

1.21665 

1.27628 

1.33822 

1.40255 

5 

6 

1.19405 

1.26531 

1.34009 

1.41851 

1.50073 

6 

7 

1  22987 

1.31593 

1.40710 

1.. 50363 

1.60578 

7 

8 

1.26677 

1.36856 

1.47745 

1.59384 

1.71818 

8 

9 

1.30477 

1.42331 

1.55132 

1.68947 

1.83S45 

9 

10 

1.34391 

1.48028 

1.62SS9 

1.79081 

1.96715 

10 

II 

1.38423 

1.53945 

1.71033 

1.89829 

2.10485 

11 

12 

1.42.576 

1.60103 

1.79585 

2.01219 

2.25219 

12 

13 

146853 

I  66507 

1.88564 

2 13292 

2.40984 

13 

14 

1.51258 

1.73167 

1.97993 

2.20090 

2.57853 

14 

15 

1.5.5796 

1.80094 

2.07892 

2.39055 

2  75903 

15 

16 

1.60470 

1.87293 

2.18287 

2.  ,54035 

2.95216 

16 

17 

1.65284 

1.94790 

2  29201 

2.69277 

3.1,5881 

17 

18 

1.70243 

2.02581 

2.40061 

2  85433 

337993 

18 

19 

1.75350 

2.10684 

2.52695 

3.02.559 

3.61652 

19 

20 

1.80611 

2.19112 

2.65329 

3  20713  3  86968 

20 

Note. — When  there  are  months  and  days  in  the  time,  find  tho 
amount  for  the  ycarSj  and  on  this  amount  cast  the  interest  for  tho 
months  and  days  :  this,  added  to  the  last  amount,  v.-ill  be  the  re- 
quired amomit  for  the  whole  time. 

9.  What  will  be  the  compound  interest  of  §75439,75,  for  4 
years?  at  4^  per  cent  ? 

10.  What  will  SG50  amount  to  in  12  year.?,  at  3  per  cent, 
compound  intercut  ? 

1 1.  The  slave  population  in  the  United  States  and  territories 
in  1850,  was  3204318;  if  the  increase  is  5  percent  a  year, 
what  will  be  the  entire  slave  population  in  1870  ? 

12.  Wliat  would  be  the  compound  interest  of  §540,50,  at  6 
per  cent,  for  3  years  G  months  and  15  days?     (Soe  Note). 


DISCOUNT.  235 

13.  What  will  $75  amount  to  in  10  years  4  months  and  21 
days,  at  7  j^er  cent,  compound  intei'est  ? 

14.  What  will  be  the  compound  interest  of  $200,  for  1  year 
7  months  and  9  days,  at  5  per  cent  ? 

15.  A  gives  B  a  note  of  $375,40,  April  20,  1854,  payable 
Oct.  20,  185G  :  the  interest  is  to  be  added  at  the  end  of  every  6 
months,  and  compounded  at  7  per  cent :  what  will  be  the 
amount  of  the  payment  when  due  ? 

DISCOUNT. 

228.  Discount  is  an  allowance  made  for  the  payment  of 
money  before  it  is  due. 

The  Face  of  a  note  is  the  amount  named  in  the  note. 

229.  The  PRESENT  value  of  a  note  is  sucli  a  sum  as  being 
put  at  interest  until  the  note  becomes  due,  would  increase  to  an 
amount  equal  to  the  face  of  the  note. 

230.  The  DISCOUNT  on  a  note  is  the  difference  between  the 
face  of  the  note  and  its  present  value. 

1.  I  give  Mr.  Wilson  my  note  for  SlOG,  payable  in  1  year: 
what  is  the  present  value  of  the  note,  if  the  interest  is  6  per  cent  ? 
what  is  the  discount  ? 

PROPORTION. 

1  +  its  interest     :     $1     :  :     given  sum     :     its  present  value. 

OPERATION. 

Analysis. — Since  1  dolJar  in  1  year,  SlOG -f- 1.06  =  $100 
at  G  per  cent,  will  amount  to  81,06.  the 

pre.'^ent  value  will  be  as  many  dollars  as  proof. 

$LOG   is  contained  times  in  the  face  of  Int.  SlOO  lyr.  =  $     6 

the  note;  viz..  ?plOO;  and  the  discount  Principal  =    100 

will  be  SlOG  —  100  =  $6  :  hence,  Amount    SlOG 

Discount         6 

228.  What  is  discount  1     What  is  the  face  of  a  note  1 

229.  What  is  present  value  1 

230.  What  is  the  discount  ? 


236  PERCENTAGE. 

Rule. — Divide  the  face  of  the  note  by  1  dollar  plus  the  in. 
terest  of  1  dollar  for  the  given  time,  and  the  quotient  will  be  the 
present  value. 

Note. — When  payments  are  to  be  made  at  different  i\me,Sy  find  the 
present  value  of  the  sums  separately,  and  their  sum  will  be  the  present 
value  of  the  note. 

EXAMPLES. 

Ul.  What  is  the  present  value  of  a  note  of  S615,  due  1  year 
4  months  hence,  at  7  per  cent  ? 

2.  "Wliat  is  the  present  value  of  $202,58,  clue  in  1  year  7 
months  and  18  days,  at  G  per  cent  ? 

3.  How  much  should  I  deduct  for  the  present  payment  of  a 
note  of  $721,  due  in  7  months  and  6  days,  at  5  per  cent? 

4.  If  a  note  for  $5160  is  payable  Feb.  4th,  1857,  what  is  its 
value  Sept.  lOth,  185G,  interest  being  reckoned  at  8  per  cent  ? 

5.  "What  sum  of  money  will  amount  to  $2500,  in  2  years  7 
months  and  12  days,  at  12  per  cent  ? 

G.  "What  is  the  present  value  and  discount  of  $3000,  paya- 
ble in  1  year  2  months  and  20  days,  at  7  per  cent  ? 

7.  Bouglit  property  to  the  amount  of  $5000,  and  agreed  to 
pay  for  it  in  four  equal  installments ;  the  first  in  3  months ;  the 
second,  in  6  months ;  the  third,  in  9  montlis ;  and  the  fourth,  in 
1  year :  how  much  cash  will  discharge  the  debt  ?  Interest  6 
per  cent.  ? 

~  8.  If  I  loan  a  sum  of  money  on  interest  at  G]-  per  cent ,  for 
7  months  and  15  days,  and  at  the  end  of  the  time  receive 
$4987,50  for  principal  and  interest,  how  nnicli  did  I  loan  ? 
;>9.  A  held  a  note  of  81400  against  B,  payable  Aug.  1st, 
185G  ;  B  paid  it  May  15th,  1856  :  what  sum  did  he  pay,  the 
inlerest  being  7  per  cent.  ? 

10.  A  Hour  merchant  bought  for  cash  300  barrels  of  flour,  for 
^10,50  per  barrel ;  he  sold  it  the  same  day  for  Si  2  a  barrel, 
and  took  a  note  at  3  moijths  :  what  was  the  cash  value  of  tlie 
Bale,  and  wliat  his  gain  il'the  interest  is  reckoned  at  7  per  cent.  ? 

11.  A  man  purcliased  a  house  and  lot  for  ^lOOOO,  on  tlio 
following  terms :  $5000   in   cash,  $2500  in  3  mouths,  and  (lie 


BANKING.  237 

balance  in  G  months :  what  was  the  cash  value  of  the  property, 
interest  being  reckoned  at  6  per  cent.  ? 

12.  A  provision  dealer  bought  for  cash  78  firkins  of  butter, 
of  8G  pounds  each,  at  25  cents  per  pound.  He  sold  it  imme- 
diately at  25-^  cents  per  pound,  and  took  a  note  for  tlie  amount 
at  4  months  :  did  he  gain  or  lose,  and  how  much  ;  interest  be- 
ing reckoned  at  8  per  cent.  ? 

13.  Which  is  the  more  advantageous,  to  buy  sugar  at  7-i 
cents  a  pound,  on  4  months,  or  at  8  cents  a  pound  on  G  months, 
at  6  per  cent,  interest  ? 

/114.  Bought  land  at  $10  an  acre  :  what  must  I  ask  per  acre  if  I 
abate  10  per  ct.,  and  still  make  20  per  ct  on  the  purchase  monc}-  ? 
15.  A  merchant  owes  three  notes,  viz.,  $1000  pa3able  Aug. 
1st,  1855  ;  -S-oOO.  payable  Oct.  lOtb.  1855.  and  SOOO  payal)lc' 
Nov.  1st,  1855:  what  is-thecash  value  of  the  three  notes,  July 
1st,  1855,  reckoning  interest  at  G  per  cent. ;  and  what  is  the 
difference  between  that  value  and  their  amounts  at  the  times 
Avhen  they  fall  due,  if  interest  be  reckoned  from  July  1st. 

BANKING. 

231.  Banks  are  corporations  created  by  law  for  the  purpose 
of  receiving  deposits,  loaning  money,  and  furnishing  a  paper 
ciiTulation  I'epresented  by  specie. 

The  notes  made  by  a  bank  circulate  as  money,  because  they 
are  payable  in  specie  on  presentation  at  the  bank.  They  are 
called  bank  notes,  or  bank  bills. 

The  note  of  an  individual,  or  as  it  is  generally  called,  a 
promissoiy  note,  or  note  of  hand,  is  a  positive  engagement,  in 
writing,  to  pay  a  given  sum,  either  on  demand  or  at  a  specified 
time. 

FORMS    OF   NOTES. 

Negotiable  Note. 

$25,50.  Pi-ovidence,  May  1,  185G. 

For  value  received  I  promise  to  pay  on  demand,  to 
Abel  Bond,  or  order,  twenty-five  dollars  and  fifty  cents. 

Reuben  Holmes. 


233  PERCENTAGE. 

Note  Payable  to  Bearer. 


No.  2. 


$875,39.  St.  Louis,  May  1,  1855. 

For  value  received  I  promise  to  pay,  six  months  after 
date,  to  John  Johns,  or  bearer,  eight  hundred  and  seventy-five 
dollars  and  thirty-nine  cents.  Pierce  Penny. 


Note  by  two  Person fi. 
No.  3. 


$659,27  Bufialo,  June  2,  1856. 

For  value  received  we,  jointly  and  severallj^,  promise 
to  pay  to  Richard  Ricks,  or  order,  on  demand,  six  hundred  and 
fifty-nine  dollars  and  twenty-seven  cents.  Enos  Allan. 

John  Allan. 


Note  Payable  at  a  Bank. 
No.  4. 


$20,25.  Chicago,  May  7,  1856. 

Sixty  days  after  date  I  promise  to  pay  John  Anderson, 
or  order,  at  the  Bank  of  Commerce  in  the  city  of  New  York, 
twenty  dollars  and  twenty-five  cents,  for  value  received. 

Jesse  Stokes. 

remarks  relating  to  notes. 

1.  The  person  who  si^nis  a  note,  is  called  the  drawer  or  malrr  of 
the  note;  thus,  Reuben  Holmes  is  the  drawer  of  note  No.  1. 

2.  The  person  who  has  the  rightful  pos.scssion  of  a  note,  is  called 
ihe  holder  of  the  note. 

3.  A  note  is  said  to  be  negotiable  -when  it  is  made  payable  to  A  B, 
or  order,  who  is  called  the  payee  (see  No.  1).  Now,  if  Abel  Bond, 
to  whom  this  note  is  mndo  payable,  writes  liis  name  on  the  back  of 
it,  he  is  said  to  endorse  the  note,  and  he  is  called  the  endorser;  and 
when  the  note  becomes  due,  the  holder  nnist  first  demand  payment 
of  till'  maker,  Reuben  Holmes,  and  if  lie  declines  paying  it,  tlio 
holder  may  then  require  payment  of  Abel  Bond,  the  endorser. 


231.    Wliat  arc  Banks'!      M'liy  do  (ho  notes  of  l)nnks  circulate  at 

nioni'V  !      W'li.il  are  tlicy  called  '      W  h^t  is  a  promissory  note! 


BANKING.  239 

4.  If  the  note  is  made  payable  to  A  B,  or  bearer,  then  the  drawer 

alone  is  responsible,  and  lie  must  pay  to  any  person  who  holds  the 
note. 

5.  The  time  at  which  a  note  is  to  be  paid  should  always  be 
named,  but  if  no  time  is  specified,  the  drawer  must  pay  when  re- 
quired to  do  so,  and  the  note  will  draw  interest  after  the  payment  is 
demanded. 

6.  AYhen  a  note,  payable  at  a  future  day,  becomes  due,  it  will 
draw  interest,  though  no  mention  is  made  of  interest. 

7.  In  each  of  the  States  there  is  a  rate  of  interest  established  by 
law,  which  is  called  the  legal  interest,  and  when  no  rate  is  specified, 
the  note  will  always  draw  legal  interest.  If  a  rate  hi"-Ler  than 
legal  interest  is  named  in  the  note,  or  agreed  upon,  the  Cr<?  Aer,  in 
most  of  the  States,  is  not  bound  to  pay  the  note. 

8.  If  two  persons  jointly  and  severally  give  their  not%  (see  No.  3). 
it  may  be  collected  of  either  of  them. 

9.  The  words  ''For  value  received,"  should  be  ex^rer,sed  in  every 
note. 

10.  When  a  note  is  given,  payable  on  a  fixed  dr./;  and  in  a  specific 
article,  as  in  wheat  or  rye,  payment  must  be  vjfiered  at  the  spe- 
cified time,  and  if  it  is  not,  the  holder  can  d-imand  the  value  in 
money. 

11.  D.iYS  OF  GRACE  are  days  allowed  for  (he  payment  of  a  note 
after  the  expiration  of  the  time  named  on  its  face.  By  mercantile 
usage,  a  note  does  not  legally  fall  due  until  3  days  after  the  expira- 
tion  of  the  time  named  on  its  face,  unless  the  note  specifies  "  ivithout 
graceP  For  example,  No.  2  would  be  due  on  the  4th  of  November, 
and  the  three  additional  days  are  called  days  of  grace. 

When  the  last  day  of  grace  happens  to  be  a  Sunday,  or  a  holiday, 
such  as  New  Ycar'.s  day  or  the  4th  of  July,  the  note  must  be  paid  the 
day  before  ;  that  is,  on  the  second  day  of  grace. 

12.  There  are  two  kinds  of  notes  discounted  at  banks  :  1st.  Notes 
given  by  one  individual  to  another  for  property  actually  sold — these 
are  called  business  notes,  or  btismess  paper.  2d.  Notes  made  for  the 
purpose  of  borrowing  money,  which  are  called  accommodation  nutes, 
or  accommodation  paper.  The  first  class  of  paper  is  much  preferred 
by  the  banks,  as  more  likely  to  be  paid  when  it  falls  due,  or  in  mer- 
cantile phrase,  "when  it  comes  to  maturity." 


240  PERCENTAGE. 

BANK  DISCOUNT. 

232.  Bank  Discount  is  the  charge  made  by  a  bank  for  the 
payment  of  money  on  a  note  before  it  becomes  due. 

By  the  custom  of  banks,  this  discount  is  the  interest  on  the 
amount  named  in  a  note,  calculated  from  the  time  the  note  is 
discounted  to  the  time  when  it  falls  due ;  in  which  time  the 
three  days  of  grace-are  always  included  (see  remark  11).  The 
interest  on  notes  discounted  at  bank  is  always  paiCi?  in  advance. 

X?uLE. — Add  3  days  to  (he  time  which  (he  no(e  has  to  ruHf 
and  (hen  calculate  (he  in(ercst  for  (hat  time,  a(  (he  given  ra(e. 

EXAMPLES. 

1.  What  is  the  bank  discount  on  a  note  of  $300,  for  4  months, 
at  6  per  cent  per  annum  ? 

2.  What  is  the  bank  discount  on  a  note  of  $200,  payable  in 
5  months,  at  9  per  cent  ? 

3.  \Yhat  is  the  bank  discount  and  proceeds  of  a  note  of  $500, 
at  G^  per  cent.,  payable  in  Q\  montlis  ? 

4.  What  is  the  cash  value  of  a  note,  payable  at  bank,  of 
$1255,38,  payable  in  4  montlis,  at  7  per  cent.  ? 

5.  Wiiat  would  be  the  bank  discount  on  a  note  of  8500,  due 
Aug.  13th,  1855,  and  discounted  July  1st,  1855,  reckoning 
interest  at  7  per  cent.  ? 

6.  I  bought  4368  bushels  of  wheat  at  $1,25  a  bushel,  and 
Eold  it  the  same  day  for  $1,30  a  bushel  on  a  note  of  4  months. 
If  I  get  this  note  discounted,  at  bank,  at  7  per  cent,  wluit  do  I 
gain  or  lose  ? 

7.  What  is  the  difference  between  tlie  true  and  bank  discount, 
on  $7000,  payable  in  7  montlis,  at  G  per  cent  ? 

8.  What  is  the  dilference  between  the  true  and  bank  discount, 
of  $10000,  payable  in  4i  months,  at  8  per  cent? 

9.  January  1st,  1855,  a  note  was  given  for  -i^lOOO,  at  5^  per 
cent.,  to  be  paid  May  1st,  next  following:  what  was  its  cash 
value  at  bank  ? 

232.  What  is  bank  discount  ?  How  is  interest  calculated  by  the  custom 
of  banks  1      When  Ik  tho  intrrc^^  paid  '      Mow  do  you  find  tlic  interest.  ? 


NOTES  OF  kt:qciki!:d  values.        241 

NOTES  OF  KEQUIKED  VALUES. 

233. — !•  For  what  sum  must  a  note  be  drawn  at  4  months 
and  12  days,  so  that,  when  discounted  at  bank,  at  G  per  cent, 
the  amount  received  shall  be  $400  ? 

Analysis. — If  we  find  the  interest  on  1  dollcir  for  the  given  time, 
and  then  subtract  that  interest  from  1  dollar,  the  remainder  will  be 
Ihe  present  value  of  1  dollar,  due  at  the  expiration  of  that  time. 
Then,  the  number  of  times  Avhich  the  present  value  of  the  note  eon- 
tains  the  present  value  of  1  dollar,  will  be  the  number  of  dollars  for 
which  tlie  note  must  be  drawn  ;  hence, 

Divide  ihe  2^rescnt  value  of  the  note  hy  the  'present  value  of 
1  dollar,  reckoned  for  ihe  same  time  and  at  the  same  rate  of 
interest,  and  the  quotient  xoill  he  the  face  of  the  note. 

OPERATION. 

Interest  of  $1  for  the  time  4mo.  12da.  +  3da.  of  grace  =  Amo.  loda, 
=  $0.0225,  which  taken  from  Si,  gives  present  value  of  ^1  =5^0.9775  ; 
then,  $400 -f- .09775  =  $409,207  +  =  face  of  note. 

PROOF. 

Interest  on  $409,207  for  4  months  and  15  days,  at  6  per  cent,  is 
S9,207  +,  which  being  taken  from  the  face  of  the  note,  leaves  $400, 
its  present  bank  value. 

2.  For  what  sum  must  a  note  be  drawn  at  7  per  cent,  paya- 
ble in  G  months,  so  that  when  discounted  at  a  bank  it  shall  pro- 
duce $285,95  ? 

3.  How  large  a  note  must  I  make  at  a  bank,  at  G  per  cent, 
payable  in  G  months  and  9  days,  to  produce  $674,89  ? 

4.  A  holds  a  note  against  B  for  $1500,  to  run  G  months  from 
Aug.  1st,  without  interest.  Oct.  1st,  he  wishes  to  pay  a  debt  at 
the  bank  of  $1000,  and  turns  in  the  note  at  a  discount  of  5  per 
cent  in  payment :  how  much  must  he  receive  back  from  the 
hank  ? 

'Z'.n.    liow  do  you  find  ihe  face  of  a  ncjte  of  a  rctjiiinnl  pronout  value  1 


242  COMMISSION, 

5.  Marsh,  Dean  &  Co.,  purchase  of  John  Jones  380  barrels 
uf  flour,  at  -$9,12L  a  barrel,  for  -which  they  give  him  a  bank 
note  for  90  days,  at  6  per  cent,  for  such  a  sum  that  if  discounted 
he  shall  receive  the  above  price  for  his  flour :  what  was  the 
face  of  the  note  ? 

COMMISSION. 

234.  Commission  is  an  allowance  made  to  an  a^ent  for 
buying  or  selling,  and  is  reckoned  at  a  certain  rate  per  cent  on 
the  money  used  in  the  purchase  or  sale. 

The  commission  for  the  purchase  or  sale  of  goods,  in  the  city 
of  New  York,  varies  from  2i  to  121  per  cent,  and,  under  some 
circumstances,  even  higher  rates  are  paid.  For  the  sale  of  i*eal 
estate,  the  rates  are  lower,  varying  from  one-quarter  to  2  per 
cent. 

EXAMPLES. 

1.  A  commission  merchant  sold  a  lot  of  goods  for  which  he 
received  $7540  ;  he  charged  2^  per  cent  commission :  what 
was  the  amount  of  his  commission,  and  Iiow  nmch  must  he  pay 
over  ? 

OPERATION. 

Analysis. — We  find  the  commission  S7540 

as  in  simple  percentai^e,  by  multiply-  .02i^ 

ing  by  the  decimal  which  expresses  the  $188,50  commission, 
rate.     The  principal,  diminished  by  the  $7540 

commission,  gives  the  amount  to  be  paid  1 88.50 

over.  $7351,50  amt.  pd.  over. 

Rule. — Multiply  the  amount  invested  by  the  rate  expressed 
decimnlly,  and  the  product  will  be  the  commission. 

2.  A  commission  merchant  receives  $1399,77,  to  be  invested 
in  groceries  ;  he  is  to  receive  3  per  cent  commission  on  the 
amount  of  the  purcliase  :  what  amount  is  laid  out  in  groceries  ? 


234.  What  is  con)mis.sion  ?  Whrit  arc  the  |;eneral  rales  of  cominission 
in  the  city  of  New  York  !  How  do  you  find  tlic  amount  of  commission 
on  a  given  sum  !  How  do  you  find  the  amount  to  tic  invcstrd  c\clu.sivo 
of  the  rijnimisbion  ? 


EXAMPLES.  243 

Analysis. — Since  the  broker  receives  operation. 

3  per  cent,  it  will  require  $1.03  to  pur-       1.03)  1399,77  ($1359 
chase  1  dollar's  worth  of  stock ;  hence, 

there  will  be  as  many  dollars  Avorth  purchased  as  $1.03  is  contained 
times  in  $1399,77;  that  is,  $1359. 

Rule. — Divide  the  amount  to  be  expended  by  $1  plus  the 
commission,  and  the  quotient  ivill  be  the  amount  exclusive  of  the 
commission, 

3.  An  auctioneer  sold  a  house  for  $3125,  and  the  furniture  for 
$1520  :  what  was  his  commission  at  f  per  cent  ? 

4.  A  flour  merchant  sold  on  commission  750  barrels  of 
flour  at  $9,75  a  barrel :  what  was  his  commission  at  2^  per 
cent  ? 

5.  Sold  at  auction  96  ho2;sliGads  of  su2:ar,  each  weif^hino; 
Sicirt.  and  50//^,  at  $6,50  per  hundred :  what  was  the  auctioneer's 
commission  at  1|-  per  cent,  and  to  how  much  Avas  the  owner 
entitled  ? 

6.  An  agent  purchased  2340  bushels  of  wheat  at  $1,75  a 
bushel,  and  charged  2f  per  cent  for  buying,  If  per  cent  for 
shipping,  and  the  freight  cost  2  per  cent :  what  was  his  com- 
mission, and  what  did  tlie  wheat  cost  the  owner  ? 

7.  A  town  collector  receives  4i  per  cent  for  collecting  a  tax 
of  82564,250  :  what  was  the  amount  of  his  percentage  ? 

8.  A  bank  fails,  and  has  in  circulation  bills  to  the  amount  of 
$267581.  It  can  pay  9i  per  cent :  how  much  money  is  there 
on  hand  ? 

9.  I  paid  an  attorney  6|-  per  cent  for  collecting  a  debt  of 
§7320,25  :  how  much  did  I  receive  ? 

10.  My  commission  merchant  sells  goods  to  the  amount  of 
Si 000,  on  which  I  allow  him  5  per  cent,  but  as  he  pays  over 
the  money  before  it  comes  due,  I  allow  him  li-  per  cent,  mo  e  : 
how  much  am  I  to  receive  ? 

11.  I  forward  $2608,625  to  a  commission  merchant  in 
Chicago,  requesting  him  to  purchase  a  quantity  of  corn  ;  h^  is 
to  receive   2^  per  cent,  on  the   purchase :  what  does   his  com 


24:4  COMMISSION. 

mission  amount   to,  and   how  much  corn  can  he  buy  -with  the 
remainder,  at  56  cents  a  bushel? 

12.  I  am  obliged  to  sell  S2640  in  bills  on  the  bank  of  Dela- 
ware, upon  -which  there  is  a  discount  of  2|  per  cent :  how  much 
bankable  money  will  I  receive  after  deducting  the  brokerage, 
which  is  ^  per  cent  ? 

13.  My  agent  at  Havana  purchased  for  me  a  quantity  of 
sugar  at  61  cents  a  pound,  for  which  I  allow  him  a  commission 
of  Ij  per  cent.  His  commission  amounts  to  ^42,66 :  how 
many  barrels  of  sugar  of  240  pounds  each  did  he  purchase,  and 
how  much  money  must  I  send  him  to  pay  for  it,  including  his 
commission  ? 

14.  A  dairyman  sent  an  agent  3476  pounds  of  cheese,  and 
allows  him  3^  per  cent  for  selling  it :  how  much  would  he  receive 
after  deducting  the  commission,  if  it  were  sold  for  122-  cents 
per  pound  ? 

15.  A  factor  receives  $708,75,  and  is  directed  to  purchase 
iron  at  $45  a  ton  ;  he  is  to  receive  5  per  cent  on  the  money 
paid  :  how  much  iron  can  he  purchase  ? 

IP  A  person  having  $1500  in  bills  of  the  State  Bank  of 
Indiana,  upon  which  there  is  a  discount  of  2^  per  cent,  and 
$1000  of  the  Bank  of  Maryland,  upon  which  thei^e  is  a  dis- 
count of  31  per  cent :  what  will  be  the  loss  in  changing  the 
amount  into  current  money  ? 

*17.  I  sent  $2204  to  a  ft-iend,  to  ])urcliase  for  me  u  number 
of  shares  in  the  Illinois  Central  Railroad ;  after  deducting  his 
commission  of  '{  per  cent.,  how  many  shares  can  he  buy  at 
$109,3735  a  share  ? 

18.  A  gentleman  sent  his  son  §56448,90,  to  invest  in  govern- 
ment C\\ud<  ;  after  deducting  the  brokerage  of  f  of  2  per  cent. 

■f hat  sum  did  he  invest  ? 


*  Sec  Art.  240.   Paye  246. 


STOCKS    AND    BKOKEKAGE.  245 

STOCKS  AND  BROKERAGE. 

235.  A  Corporation  is  a  collection  of  i^ersons  authorized 
i)j  law  to  do  business  together.  The  law  Avhich  defines  their 
rights  and  powers  is  called  a  Charter. 

236.  Capital, or  Stock,  is  the  money  paid  in  to  carry  on 
the  business  of  the  corporation,  and  the  individuals  so  con- 
tributing are  called  Stockholders.  This  capital  is  divided  into 
equal  parts  called  Shares,  and  the  written  evidences  of  owner- 
ship are  called  Certificates. 

237.  When  the  United  States  Government,  or  any  of  the 
States,  borrows  money,  an  acknowledgment  is  given  to  the 
lender,  in  the  form  of  a  bond,  bearing  a  fixed  interest.  Such 
bonds  ai-e  called  United  States  Stock,  or  State  Stock. 

238.  The  PAR  VALUE  of  stock  is  the  number  of  dollars 
named  in  each  share,  generally  100.  The  market  value  is 
what  the  stock  brings  per  share  when  sold  for  cash. 

If  the  market  value  is  above  the  par  value,  the  stock  is  said 
to  be  at  a  premium ;  but  if  the  market  value  is  below  the  par 
value,  it  is  said  to  be  at  a  discount. 

Let  1  =  par  value  of  1  dollar ;  then, 

1  +  premium  =  market  value  of  1  dollar,  Avhen  above 

par : 
1  —  discount  =  market  value  of  1  dollar,  when  below 

par. 

239.  A  DIVIDEND  is  an  interest  or  profit  on  a  stock,  and  is 
estimated  at  so  much  per  cent  on  the  par  value  of  a  share. 

235.  What  is  a  corporation'!     What  is  a  charter! 

236.  What  is  capital  or  stock  !     AMiat  are  shares  ? 

237.  What  are  TInited  States  Stocks  '     What  are  State  Stocks  1 

238.  What  is  the  par  value  of  a  stock  1  What  is  the  market  value  1 
If  the  market  is  above  the  par  value,  what  is  said  of  the  stock  ?  If  it  is 
below,  what  is  said  of  the  stock  ?  What  is  the  market  value  when  above 
pari     What  when  below  1 

239.  What  is  a  dividend  ''     On  what  i.-i  it  estimated  ! 


24(3  pi<:kci':ntagj':. 

240.  Brokeuage  is  an  allowance  made  to  an  agent  wlio 
buys  or  sells  stocks,  uncurrent  money,  or  bills  of  exchange^ 
and  is  generally  reckoned  at  so  much  per  cent  on  tlie  pur  value 
of  the  stock.  The  brokerage,  in  the  city  of  New  York,  is 
generally  one-fourth  per  cent  on  the  par  value  of  the  stock. 

241.  To  find  the  value  of  stock  which  is  above  or  below  par. 

1.  What  is  the  value  of  $5000  of  stock,  reckoned  at  par, 
when  the  stock  is  at  a  premium  of  9  per  cent  ? 

Analysis. — The    question    is,    how   much  operation. 

money  ^Yill  it  take  to  purchase  $1  of  stock,  at  $5600 

par.     If  the  stock  is  above  par.  it  will  take  1  1.09 

dollar  plus  the  premium  :  if  it  is  below  par,  it  50400 

■will  take  1  dollar  minus  the  premium  ;  hence,  5600 

$6104.00 

Multijoly  the  cost  of  the  stock,  at  'par,  ly  the  price  ofl  dollar 
of  stock,  expressed  decimally,  and  the  product  will  be  the  value. 

Note. — When  there  is  a  charge  for  brokerage,  it  must  be  added  to 
the  cost  of  1  dollar  of  the  stock. 

EXAMPLES. 

1.  What  is  the  cost  of  56  shares  of  New  York  Central  Rail- 
road stock,  at  51  per  cent  below  par,  the  shares  being  §100 
each,  and  the  brokerage  ^  per  cent  ? 

2.  I  bought  36  shai'es  of  $100  each,  in  the  Pennsylvania 
Coal  Company,  at  a  discount  of  12^^  per  cent,  and  sold  them  at 
a  premium  of  7  per  cent,  paying  i  per  cent  brokerage  in  each 
case  :  how  much  did  I  make  by  the  operation  ? 

3.  Tlie  par  value  of  257  shares  of  bank  stock  was  $200  a 
share :  what  is  the  present  value  of  all  the  shares,  tlie  stock 
being  at  a  premium  of  $15  per  cent  ? 

4.  AYhat  is  the  value  of  120  shares  of  Exchan2;e  Bank  Stock 
it  being  at  a  premium  of  18|-  per  cent,  and  the  par  value  be' 
ing  $150  a  share  ? 


240.  What  is  brokorafje  ^ 

241.  How  do  you  find  the  value  of  a  stuck  which  is  above  par  ! 


BTOCKS    AND   HKOKEEAGE.  2i7 

f).  What  is  the  cost  of  69  shares  of  Panama  Railroad  Stock, 
it  being  at  a  discount  of  8  per  cent,  and  tlie  par  value  being 


$125,  and  the  charge  for  brokerage  f  per  cent  ? 

6.  Gilbert  &  Co.  buy  for  Mr.  A.  200  shares  of  United  States 
Stock,  at  a  premium  of  6^  per  cent,  and  charge  i-  per  cent 
brokerage  :  if  the  shares  are  $1000  each,  how  much  money 
does  A  pay  for  the  stock  ? 

7.  Mr.  B.  bought  125  shares  of  stock  in  the  American  Guano 
Company,  at  par,  the  shares  being  $20  each.  At  the  end  of  4 
months,  he  received  a  dividend  of  5  per  cent,  and  at  the  end 
of  10  months,  a  second  dividend  of  4  per  cent.  At  the  end  of 
the  year,  he  sold  his  stock  at  a  premium  of  10  per  cent :  how 
much  did  he  make  by  the  operation,  reckoning  the  mterest  of 
money  at  7  per  cent  ? 

242.  To  Jind  how  much  stock,  at  2)a)-  value,  a  given  sum  of 
money  tnll  purchase,  when  the  stock  is  at  a  2'>remium  or  discount. 

1.  What  value  of  stock,  at  par,  can  be  purchased  by  $3045,38 
if  the  stock  is  at  a  premium  of  10  per  cent,  and  -^  per  cent  is 
charged  for  brokerage  ? 

Analysis. — Since  the  stock  is  at  a  premium  of  10  per  cent,  and 
the  charge  for  brokerage  is  \  per 

cent,  it  will  require  $1,105  to  pur-  operation. 

chase  $1  of  stock,  at  par  value;        1.105)3045.3S($2756  Ans. 
hence,  $3045.38  will  purchase  as 

many  dollars,  at  par,  as  $1,105  is  contained  times  in  $3045,38  ;  viz., 
$2756. 

Rule. — Divide  the  given  sum  hy  the  cost  of  $1  of  the  stock, 
expressed  decimally,  and  the  quotient  u'ill  denote  the  par  value 
of  the  stock  p)urchased. 

EXAMPLES. 

1.  A  person  wishes  to  invest  $3000  in  bank  stock,  which  ia 
at  a  discount  of  15  per  cent :  what  amount,  at  par  value,  can 
he  purchase  ? 


242.  How  do  you  find  the  sum  which  will  purchase  a  given  amount  of 
stock -at  par  value  l 


2-18  PERCENTAGE. 

2.  How  many  shares  of  Galena  and  Chicago  Raih'oad  Stock 
can  be  bought  for  $6384,  at  an  advance  of  14  per  cent  on  tne 
par  value  of  $100  a  share  ? 

S.  When  bank  stock  sells  at  a  discount  of  7\  per  cent,  Avhat 
amount  of  stock,  at  par  value  will  §3700  buy  ? 

4.  A  person  has  $7000,  which  he  wishes  to  invest :  what  v.ill 
it  purchase  in  5  per  cent  stocks,  at  a  discount  of  31  per  cent,  if 
he  pays  i  per  cent  brokerage  ? 

5.  How  much  6  per  cent  stock,  at  par,  can  be  purchased  for 
$8700,  at  81  per  cent  premium,  i  per  cent  being  paid  to  the 
broker  ? 

6.  A  person  owning  $12000  in  government  funds,  desires  to 
purchase  stock  in  the  American  Exchange  Bank.  The  funds 
are  at  a  discount  of  3^  per  cent,  while  the  bank  stock  is  at  a 
premium  of  lOi  per  cent :  what  amount  of  stock,  at  par  value, 
can  he  purchase,  allowing  the  broker's  charges  for  the  pur- 
chase to  be  ^  per  cent  ? 

243.  1st.  To  find  the  rate  of  interest  on  an  investment,  when 
the  stock  is  above  or  below  par.  2d.  To  find  how  much  the 
stock  must  be  above  or  belotv  par  to  irroduce  a  given  rate. 

1.  What  is  the  rate  of  interest  on  an  investment  in  G  per 
cent  stocks,  when  they  are  at  a  discount  of  25  per  cent  ? 

Analysis. — The  interest  on  the  slock   is 
computed  on  its  par  value,  while  the  interest  operation. 

on  the  investment  is  computed  on  the  amount  •'  5 

paid  :  hence,  1   dollar  multiplied  by  the  iu- 


.06 


terest.  ou  the  stock  will  be  equal  to  the  cost  x  =  .08 

of  ]  dollar  multiplied  by  the  interest  on  the        Ans.  8  per  cent. 

investment. 

Rule. — I.  Divide  1  dollar  multiplied  hj  the  interest  on  $1, 
by  (lie  price  of  $1  of  the  slock,  and  the  quotient  tvill  be  the 
rale. 


243.  How  do  you  find  the  rate  of  interest  on  an  investment  when  the 
stock  is  above  or  below  par  1  How  do  you  find  the  value  of  the  stock 
whet  the  rite  is  given  1 


STOCKS    AND   BROKERAGE.  2  1:9 

II.  Divide  1  dollar  multiplied  by  the  interest  on  the  stock  by 
the  given  rate,  and  the  quotient  will  he  the  ijrice  of  $1  of  the 
stock. 

EXAMPLES. 

1.  If  I  buy  7  pel  :ent  stock  at  12-|-  per  cent  discount,  what 
is  the  i-ate  per  cent  jn  the  investment  ? 

2.  At  what  rate  of  discount  must  I  invest  in  8  per  cent  stock 
in  order  to  yield  me  10  per  cent  ? 

3.  The  stock  of  the  Erie  Railroad  is  at  G2^  per  cent :  if  it 
pays  semi-annual  dividends  of  2i  per  cent,  what  would  be  the 
rate  of  interest  on  an  investment  ? 

4.  The  bonds  of  the  Illinois  Central  Railroad  Company, 
which  bear  interest  of  7  per  cent.,  are  worth  87  per  cent.,  and 
the  charge  for  brokerage  is  -|  per  cent. :  what  would  be  the 
interest  on  an  investment  in  these  funds  ? 

5.  If  the  par  value  of  a  stock  is  $100,  and  the  interest  7  per 
cent.,  what  is  the  discount  when  an  investment  yields  12  per  ct.  ? 

6.  The  stock  of  the  Hartford  and  New  Haven  Railroad  is 
at  a  premium  of  20  per  cent :  reckoning  the  interest  on  money 
at  6  per  cent,  what  Avill  be  the  interest  on  an  investment  in  this 
stock  ? 

244.    Which  is  the  best  investment? 

1.  I  invest  $1250  in  State  Stocks  bearing  an  interest  of  6 
per  cent,  and  a  premium  of  15  per  cent.  I  invest  the  same 
amount  in  State  fives  at  12  per  cent  discount :  which  will  yield 
the  larger  interest  ? 


OPERATION. 


Analysis. — Find  the  rate  of  interest  on 


1.15 

X 


n 

.06 


the  investment  (Art.  243).  ^_  q^^i  + 


244.  How  do  you  find  which  is  the  best  investment  ? 


250  FKKCENTAGE. 


.88 
Find   the  rate  of  interest  on  the  second  x 


$1 
.05 


investment   (Art.   243).      The  comparison  2  =.0568  4- 

of  these  two  rates  will  show  which  is  the  x  =  5  ^-^  pr.  ct. 

more  profitable  stock. 

2.  "Which  is  the  best  investment,  to  buy  sixes  at  par,  or 
sevens  at  107  ? 

3.  Wliich  will  yield  the  larger  profit,  8  per  cent  stock  at  a 
premium  of  20  per  cent,  or  5  per  cent  stock  at  80  per 
cent  ? 

4.  If  I  invest  $2000  in  state  stocks  at  5  per  cent,  at  par,  and 
the  same  amount  at  G  per  cent,  at  90,  what  will  be  the  diifer- 
ence  of  the  proceeds  of  the  investments  at  the  end  of  5 
years  ? 

PROFIT  AND  LOSS. 

245.  Profit  or  Loss  is  a  process  by  which  merchants  dis- 
cover the  amount  gained  or  lost  in  the  purchase  and  sale  of 
goods.  It  also  instructs  them  how  much  to  increase  or  diminish 
the  price  of  their  goods,  so  as  to  make  or  lose  so  much  per  cent. 

1.  Bought  325  bushels  of  wheat  at  $1,37  a  bushel,  and  sold 
it  at  $1,45  a  bushel :  what  was  the  profit  ? 

OPERATION. 

Analysis. — We  first  find  the  $1,45  sold  per  bushel, 

profit  on   a  single   bnslicl,   and  1,37  cost  per  bushel, 

then  multiply  by  the  number  of  ,08  profit  on  1  bushel. 

bu.shcls,  which  is  325  :  the  pro-  then,  325  x  ,08 ^$26,00  profit, 
duct  is  the  profit. 

2.  Bought  a  piece  of  broadcloth  containing  36  yards,  at  $2,75 
per  yard  ;  it  proving  damaged,  was  sold  at  a  loss  of  $18  :  for 
what  was  it  sold  per  yard  ? 


215.   What  is   Profit  or  Loss  !     How  do  you   find   the  entire  profit  ol 
loss  ?     How  do  you  find  the  profit  or  loss  on  unity  \ 


^ItOFIT    AND   LOSS.  251 

Analysis. — Find  the  cost,  which  is  operation. 

$99  ;  then  subtract  the  loss  $18,  and     $2,75  x  3G  =  $99 
then  divide  by  the  number  of  yards  :  Loss  1 8 


the  quotient  \A-ill  be  the  answer:  hence,  36)81  ($2, 25 

I.  When  the  entire  profit  or  loss  is  required : 

Find  the  iirojit  or  loss  on  unity  and  multiply  it  by  the  num- 
ber tvhich  shares  the  profit  or  loss. 

II.  When  the  profit  or  loss  on  unity  is  required  : 

Find  the  entire  profit  or  loss  and  divide  it  by  the  number 
ivhich  shares  the  profit  or  loss. 

EXAMPLES. 

1.  Bought  0  barrels  of  sugar,  each  weighing  250  pounds,  at 
7  cents  a  pound  :  how  much  profit  would  be  made  if  it  were 
sold  at  8^  cents  per  pound  ? 

2.  If  in  3  hogsheads  of  molasses  which  cost  $68,04,  one-third 
leaked  out,  what  must  the  remainder  be  sold  for  per  gallon  to 
realize  a  profit  of  $2,52  on  the  whole  ? 

3.  A  farmer  bought  a  flock  of  360  sheep  ;  their  keeping  for 
1  year  cost  $0,75  a  head ;  their  wool  was  worth  1  dollar  and 
25  cents  a  head,  and  one-fourth  of  them  had  laiTibs,  each  of 
which  was  worth  one-half  as  much  as  a  fleece  :  what  was  the 
profit  of  the  purchase  at  the  end  of  the  year  ? 

246.  Given  the  per  cent  of  the  yain  or  loss,  and  the  amount 
of  tJie  sale,  to  find  the  cost. 

1.  A  ffrocer  sold  a  lot  of  susrars  for  $477,12,  which  was  an 
advance  of  12  per  cent  on  the  cost :  what  was  the  cost  ? 

Analysis. — 1  dollar  of  the  cost  plus  12  per         opkration. 
cent;  will  be  what  that  wliich  cost  $1  sold  for,  1.12)477.12 

viz.,  $1,12  :  hence,  there  will  be  as  many  dol-  $42(3  Ans. 

lars  of  cost,  as  $1,12  is  contained  times  in  what 
the  goods  brought. 

246.  How  do  you  find  the  cost  when  you  know  the  per  cent  and  the 
amount  of  sale  1 


252  PERCENTAGE. 

2.  Mr.  A.  bought  a  lot  of  sugars,  but  finding  them  of  an  in- 
ferior quality,  sold  them  at  a  loss  of  15  per  cent,  and  found 
that  they  brought  $3-10  :  what  did  they  cost  him  ? 

Analysis. — 1  dollar  of  the  co.^t  less  15  per        operation. 
cent,  Avill  be  what  that  which  cost  1  dollar  sold  .85)340 

for,  viz..  $0.85  :  hence,  there  will  be  as  many  $400 

dollars  of  cost,  as  .85  is  contained  times  in  what 
the  goods  brought. 

Rule. — Divide  the  amount  received  by  1  plus  the  per  cent 
when,  there  is  a  gain,  and  by  1  minus  the  per  cent  when  there  is 
a  loss,  and  the  quotient  loill  be  the  cost. 

EXAMPLES. 

1.  I  sold  a  parcel  of  goods  for  $195,50,  on  which  I  made  15 
per  cent :  what  did  they  cost  me  ? 

2.  Sold  78cwt.  dqr.  14//;.  of  sugar,  at  8  cents  a  pound,  and 
gained  15  per  cent :  how  much  did  the  whole  cost  ? 

3.  A  merchant  having  a  lot  of  flour,  asked  33^  per  cent  more 
than  it  cost  him,  but  was  obliged  to  sell  it  12^  per  cent  less  than 
his  asking  price  ;  he  received  for  the  flour,  §7.015  a  barrel : 
what  did  it  cost  him  ? 

4.  A  dealer  sold  two  horses  for  $472,50  each,  and  gained  on 
one  35  per  cent,  but  lost  10  per  cent  on  the  other  :  what  was 
the  cost  of  each,  and  what  was  his  gain  ? 

2-17.  To  find  the  selling  price  of  an  article  so  os  to  gain  or 
lose  a  certain  per  cent. 

1.  A  grocer  bought  12  barrels  of  sugar,  for  which  he  paid 
§18,50  a  barrel :  what  must  he  sell  it  for  to  yield  him  a  profit 
of  15  per  cent  ? 

Analysis. — Since  the  amomit  f  »r  which  operation. 

the  sugar  is  to  be  sold  exceeds  its  cost  by  $18,50  X  12  =  $222 

15  per   cent.,  that  amount  added   to  the  15  per  cent     33. .'JO 

cost  will   give   what   it   mu&t  be  sold  for.  Ans.  $255.30 
When  tliere  i.s  a  loss,  subtract. 


PKOFIT    AND   LOSS.  255 

Rule. — Add  the  iwrcentage  to  the  cost,  token  there  is  a  gain, 
and  subtract  it  ivhen  there  is  a  loss. 

EXAMPLES. 

1.  A  farmer  sells  375  bushels  of  corn  for  75  cents  a  bushel ; 
the  purchaser  sells  it  at  an  advance  of  20  per  cent :  how  much 
a  busliel  did  he  receive  for  the  corn  ? 

2.  A  merchant  buj's  a  pipe  of  wine,  for  which  he  pays 
$322,5G,  and  he  wishes  to  sell  it  at  an  advance  of  25  per  cent : 
what  must  he  sell  it  for  per  gallon  ? 

3.  A  man  bought  3275  bushels  of  wheat,  for  which  he  paid 
S3493,33l,  but  finding  it  damaged,  is  willing  to  lose  10  per 
cent. :  what  must  he  sell  it  for  per  bushel  ? 

4.  If  a  merchant  in  selling  cloth  at  $4,70  a  yard,  loses  G  per 
cent,  on  its  cost,  for  how  much  must  he  sell  it  to  gain  14  per 
cent.  ? 

5.  If  I  purchase  two  lots  of  land  for  $150,25  each,  and  sell 
one  for  40  per  cent,  more  than  it  cost,  and  the  other  for  28  per 
cent,  less,  Avhat  is  my  gain  on  the  two  lots  ? 

6.  Bought  a  cask  of  molasses  containing  144  gallons,  at  45 
cents  a  gallon,  36  gallons  of  which  leaked  out :  at  what  price 
per  gallon  must  I  sell  the  remainder  to  gain  10  per  cent,  on 
the  cost  ? 

7.  A  person  in  Chicago  bought  3500  bushels  of  wheat,  at 
^^  1.20  a  bushel :  allowing  5  per  cent  on  the  cost,  for  risk  in 
transportation,  3  percent  for  freight,  and  2  per  cent  commission 
for  selling,  what  must  it  sell  for  per  bushel  in  New  York  that 
he  may  realize  40  per  cent  net  profit  on  the  purchase  ? 

248.    Given  the  gain  or  loss  to  find  the  iier  cent. 

1.  Bought  a  quantity  of  goods  for  $200,  and  sold  them  for 
$170  :  what  per  cent  did  I  lose  on  the  purchase  ? 


247.  How  do  you  find  the  selling  price  of  an  article  so  as  to  gain  or  loso 
a  certain  per  cent  1 

248.  How  do  you  find  the  percentage  when  you  know  the  gain  or  loss ' 

12 


254  PEKCENTAGK, 

Analysis. — The  gain  or  loss  will  opekation. 

be  equal   to  the  dilierence  between      S200-S170  ===  S30 
the  cost   and  the  amount  received  $30 -^  $200  =  .15. 

from  the  sale.  If  this  difierence  be  Ans.  15  percent, 

divided  by  the  cost,  the  quotient  will 

denote   the  per  cent  on  the  cost  •  if  it  be  divided  by  the  amount  of 
the  sale,  the  quotient  will  denote  the  per  cent  on  the  sale. 

Rule. — Divide  the  gain  or  loss  by  the  number  on  ivhich  the 
2)er  cent  is  reckoned. 

EXAMPLES. 

1.  Bought  ,1  quantity  of  goods  for  $348,50,  and  sold  tho 
same  for  $425  :  what  per  cent  did  I  make  on  the  amount 
I'eceived  ? 

2.  Bought  a  piece  of  cotton  goods  for  G  cents  a  yard,  and 
sold  it  for  71  cents  a  yard  :  what  was  my  gain  per  cent  ? 

3.  If  I  buy  rye  for  90  cents  a  bushel,  and  sell  it  for  $1,20, 
and  wheat  for  $1,12-^  a  bushel,  and  sell  it  for  $1,50  a  bushel, 
upon  which  do  I  make  the  most  per  cent  ? 

4.  If  paper  that  cost  $2  a  ream,  be  sold  for  18  cents  a  quire, 
what  is  gained  per  cent  ? 

5.  How  much  per  cent  would  be  made  upon  a  hogshead  of 
sugar  weighing  Vdciut.  3qr.  14/6.,  that  cost  $8  per  cwl,  if  sold 
at  10  cents  per  pound  ? 

6.  A  hardware  merchant  bought  45 T'.  IGcwt.  25lb.  of  iron, 
at  $75  per  ton,  and  sold  it  for  $78,50  per  ton  :  what  was  his 
whole  gain,  and  how  much  per  cent  did  he  make  ? 

7.  If  25  per  cent  be  gained  on  flour  when  sold  at  $10  a 
barrel,  Avhat  per  cent  would  be  gained  when  sold  at  $11, GO  a 
barrel  ? 

NoTK. — In  this  class  of  examples,  first  find  the  cost,  as  in  Art.  240  : 
then  find  the  gain,  or  loss,  and  then  divide  by  the  number  on  which 
the  per  cent  is  reckoned. 

8.  A  lumber  dealer  sold  25G50  feet  of  lumber  at  $19,20  a 
thousand,  and  gained  20  per  cent :  how  much  would  he  have 
gained  or  lost  had  he  sold  it  at  $15  a  thousand .'' 


INSURANCE.  255 

9  A  man  sold  his  form  for  $3881,25,  by  which  he  gained 
12^  per  cent  on  its  cost :  what  was  its  cost,  and  what  would  he 
have  gained  or  lost  per  cent  if  he  had  sold  it  for  $3277,50  ? 

10.  If  a  merchant  sell  tea  at  66  cents  a  pound,  and  gain  20 
per  cent,  how  much  would  he  gain  per  cent  if  he  sold  it  at  77 
cents  a  pound  ? 

11.  Sold  5520  bushels  of  corn  at  50  cents  a  bushel,  and  lost 
8  per  cent :  how  much  per  cent  would  have  been  gained  had  it 
been  sold  at  60  cents  a  bushel  ? 

12.  A  grocer  bought  3  hogsheads  of  sugar,  each  weighing 
14121  pounds ;  he  sold  it  at  11  cents  a  pound,  and  gained  37^ 
per  cent :  what  was  its  cost,  and  for  how  much  must  he  sell  it 
to  gain  50  per  cent  on  the  cost  ? 

INSURANCE. 

249.  Insurance  is  an  agreement,  generally  in  writing,  by 
which  individuals  or  companies  bind  themselves  to  exempt  the 
owners  of  certain  property,  such  as  ships,  goods,  houses,  &;c., 
from  loss  or  hazard. 

The  Policy  is  the  written  agreement  made  by  the  parties. 

250.  The  Base  of  insurance  is  the  value  of  the  property  in- 
sured. 

251.  Premiuji  is  the  amount  paid  by  him  who  owns  the 
property  to  those  who  insure  it,  as  a  compensation  for  their 
risk.  The  premium  is  generally  so  much  jjer  cent  on  the  pro- 
perty insured. 

252.  There  are  four  cases  which  may  arise  in  questions  of 
Insurance.  The  principles  on  which  these  cases  depend  have 
already  been  considered,  and  reference  is  made  to  the  articles. 

249.  What  is  insurance  1     What  is  a  policy  1 

250.  What  is  the  base  of  insurance  1 

251.  What  is  a  premium  1 

352.  How  many  cases  are  there  which  arise  in  insurance  1  What  are 
tliey  1 


256  I'EKCKNTAGE. 

1.  To  find  the  Premium  (Art.  216). 

2.  To  find  the  Rate  (Art.  217). 

3.  To  find  the  ba.se,  or  sum  insured  (Art.  218). 

4.  To  insure  on  both  the  base  and  premium. 

253.    To  find  the  2^'remiurii : 

1.  What  would  be  the  premium  on  a  cargo  of  goods,  valued 
at  $39854,  the  insurance  being  made  at  41-  per  cent? 

OPERATION. 

Analysis. — This   is  simply  a  case  of  finding  39854 

the  percentage  when  the  base  and  rate  are  given  .045 

(Art.  216).  $1793,430 

EXAMPLES. 

1.  What  would  be  the  premium  for  insuring  a  ship  and  cargo, 
valued  at  $147674,  at  31  per  cent  ? 

2.  What  would  be  the  insurance  on  a  ship,  valued  at  S47520 
at  ^  per  cent  ?      At  i  per  cent  ?   ' 

3.  What  would  be  the  insurance  on  a  house,  valued  at  Si  6800, 
at  1^  per  cent  ?     At  |-  per  cent  ? 

4.  A  merchant  owns  |-  of  f  of  a  ship,  valued  at  $24000,  and  in- 
sures Iiis  interest  at  2^  per  cent :  what  does  he  pay  for  his  policy  ? 

5.  WJiat  will  it  cost  to  insure  a  store  worth  $5640,  at  ^  per 
cent,  and  the  stock  Avorth  $7560,  at  -|  per  cent? 

6.  A  carriage  maker  shipped  15  carriages  worth  $425  each : 
what  must  he  pay  to  obtain  an  insurance  upon  them  at  75  cents 
on  a  hundred  dollars  ? 

7.  A  merchant  imported  liJOIihd.  of  molasses,  at  35  cents  a 
gallon  :  he  gets  it  insured  for  31  per  cent  on  the  selling  price 
of  50  cents  a  gallon  :  if  the  whole  should  be  destroyed,  and  he 
get  the  amount  of  insurance,  how  much  would  he  gain  ? 

8.  If  I  get  my  house  and  furniture,  valued  at  $3640,  insured 
at  4^  per  cent,  what  would  be  my  actual  loss  if  they  were  de- 
stroyed ? 


253.  How  (3o  you  find  the  premium  1 


LIFE    INSUliAK(!K.  257 

9.  The  ship  Astoria  is  vahied  at  $20450,  and  her  cargo  at 
$25 GOO  ;  being  bound  on  a  voyage  from  New  York  to  Canton, 
insured  $12000  on  the  vessel,  at  the  St.  Nicholas  Office,  at  2|- 
per  cent,  and  $18500  on  the  cargo,  at  the  Howard  Office,  at  3^ 
per  cent :  if  the  vessel  founder  at  sea,  what  will  be  the  loss  to 
the  owner  ? 

10.  Sliipped  from  New  York  to  the  Crimea  5000  barrels  of 
flour  worth  $10,50  a  barrel.  The  premium  paid  was  $2887,50  : 
what  was  the  rate  per  cent,  of  the  insurance  ? 

11.  Paid  $120  for  insurance  on  my  dwelling,  valued  at 
$7500  :   what  was  the  rate  per  cent.  ? 

12.  A  merchant  imported  225  pieces  of  broadcloth,  each 
piece  containing  40  yards,  at  $3,50  a  yard  :  he  paid  $1323  for 
insurance  :  what  was  the  rate  per  cent.  ? 

13.  A  merchant  paid  $1320  insurance  on  his  vessel  and 
cargo,  which  was  51  per  cent  on  the  amount  insured  :  how  much 
did  he  insure  ? 

14.  A  man  pays  $51  a  year  for  insurance  on  his  storehouse, 
at  IJ-  per  cent,  and  $126,45  on  the  contents,  at  2i  per  cent: 
what  amount  of  property  does  he  get  insured  ? 

15.  A  person  shipped  15  pianos,  valued  at  $275  each.  He 
insures  them  at  3  per  cent,  and  also  insures  the  premium  at  tho 
same  rate  :  what  insurance  must  he  pay  ? 

16.  A  store  and  its  contents  are  valued  at  $16750.  The  owner 
insures  them  at  If  per  cent.,  and  then  insures  the  premium  at 
the  same  rate  :   what  amount  of  insurance  must  he  pay  ? 

LIFE  INSURANCE. 

254.  Insurance  for  a  term  of  years,  or  for  the  entire  con- 
tinuance of  life,  is  a  contract  on  the  part  of  an  authorized  asso- 
ciation to  pay  a  certain  sum,  specified  in  the  policy  of  insurance, 
on  the  happening  of  an  e^ent  named  therein,  and  for  which  the 
association  receives  a  certain  premium,  generally  in  the  form 
of  an  annual  payment. 

'2CA.  What  is  a  life  insurance  '■ 

VA 


258  PEKCENTAGB. 

255t  To  enable  the  company  to  fix  their  premiums  at  such 
rates  as  shall  be  both  fair  to  the  insured  and  safe  to  the  asso- 
ciation, they  must  know  the  average  duration  of  life  from  any 
given  time  to  its  probable  close.  This  average  is  called  the 
"  Expectation  of  Life,"  and  is  determined  by  collecting  from 
many  sources  the  most  authentic  information  in  regard  to  the 
averwje  duration  of  life  from  any  period  named. 

If  we  take  100  infants,  some  will  die  in  infancy,  some  in 
childhood,  and  some  in  old  age.  It  has  been  found,  from  care- 
ful observation,  that  if  the  sum  of  their  ages,  after  the  last 
shall  have  died,  be  divided  by  100,  the  quotient  will  be  38.72 
very  nearly :  hence  38.72  is  said  to  be  the  "  Expectation  of 
Life"  at  infancy.  The  Carlisle  Tables,  which  are  used  in 
this  country  and  England,  show  the  "  Expectation  of  Life" 
from  1  to  100  years.  At  10  years  old  it  is  found  to  be  48.82  ; 
at  20,  41. 4G;  at  30,  it  is  34.34;  at  40,  27.G1;  at  50,  it  is 
21.11;  at  GO,  14  years;  at  70,  9.19;  at  80,  O.ol  ;  at  90, 
3.28,  and  at  100,  it  is  2.28  years. 

2.56.  From  the  above  facts,  and  the  value  of  money  (winch 
is  shown  by  the  rate  of  interest),  a  comjiany  can  calculate  with 
great  exactness  the  amount  Avhich  they  should  receive  annually, 
for  an  insurance  on  a  life  for  any  number  of  years,  or  during  its 
entire  continuance. 

Among  the  principal  life  insurance  companies  in  the  United 
States,  are  the  New  York  Life  Lisurance  and  Trust  Company, 
the  Girard  Life  Lisurance,  Annuity  and  Trust  Company  of 
Philadelphia,  and  the  Massachusetts  Hospital  Life  Insui-ance 
and  Trust  Company  of  Boston.  The  rates  of  insurance,  iu 
these  companies,  differ  but  little. 

The  PKEMiUM  for  life  insurance  is  generally  at  so  much  per 
annum  on  $100  ;  and  is  always  paid  in  advance. 

2.5.').  What  is  ncccsKary  to  enable  a  company  to  fix  their  preiniiun.s  \ 
How  is  tlie  expectation  determined  !  What  do  you  understand  by  tha 
ex|)cctation  of  life  1 

!J.OG    What  may  be  calculated  from  the  necessary  facts  1 


LIFE    INSURANCE.  259 

EXAMPLES. 

1.  A  person,  20  years  of  age,  finds  that  tlie  premium,  pei 
afiiuim,  is  $1.36  on  $100  :  what  must  he  pay  to  insure  his  lift 
for  1  year  for  $8950  ? 

2.  A  man,  aged  40  years,  wishes  to  insure  his  life  for  5  years, 
and  finds  that  the  annual  rate  is  $1.8G  for  $100  :  how  much 
premium  must  he  pay  per  annum  on  -£^12500  ? 

3.  A  person,  38  years  of  age,  obtains  an  insurance  on  lii.- 
life  for  5  years,  at  the  rate  of  ^1,75  per  annum  on  %100  :  how 
mucli  is  the  annual  premium  on  $15000? 

4  A.  person  going  to  Europe,  expecting  to  return  in  2  years, 
effects  an  insurance  on  liis  life  at  ^  of  4  per  cent,  premium  on 
$100  ;  he  insures  for  $5000  :  what  is  the  annual  premium  ? 

5.  What  will  be  the  annual  premium  for  insuring  a  person's 
life,  who  is  60  years  of  age,  for  ^2000,  at  the  rate  of  •'i;4,91  on 
$100  ? 

6.  A  person,  at  the  age  of  50  years,  obtained  an  insurance 
at  4|  per  cent,  per  annum  on  each  $100  ;  he  insured  for  $1500, 
and  died  at  the  age  of  70.  How  mucli  more  was  the  insurance 
than  tlie  payments,  without  reckoning  interest  ? 

7.  A  gentleman,  47  years  of  age,  going  to  China  as  ambassa- 
dor, obtains  an  insurance  on  his  life  for  $10000,  by  paying  a 
premium  of  $2.71  })er  annum  on  every  $100,  and  dies  at  the 
middle  of  the  third  year  :  reckoning  simple  interest  on  his  pay- 
ments at  7  per  cent,  wliat  is  gained  by  the  insurance  ? 


ENDOWMENTS. 

257.  An  Exdoavmext  is  a  certain  sum  to  be  paid  at  the 
ex])iration  of  a  given  time,  in  case  the  person  on  whose  life  it  is 
taken  shall  live  till  the  expiration  of  the  time  named. 

» 
2.57.    What  is  an  endowment  1     What  does  the  table  of  endowments 

show  ]     What  maj-  be  found  from  the  table  ". 


200 


PEIICKNTAOE. 


The  following  table  shows  the  value  of  an  endowment  pur- 
chased for  $100,  at  the  several  periods  mentioned  in  the  column 
of  ages,  the  endowment  to  be  paid  if  the  person  attains  the  age 
of  21  years. 


TABLE    OF 

ENDOWMENTS. 

.  „_              Sums  to  be  paid 
'^"''-                afil,  if  alive. 

Age. 

Sura  to  be  paid 
at  21,  if  alive. 

Age. 

Sum  to  be  paid 
at  21,  if  alivo. 

Birth,     -     -     3376,84 

5  years, 

-     $210,53 

13  years, 

-     S144.12 

3  months,  -        344,28 

6     " 

198,83 

14'  '' 

137.86 

6      "           -        331,46 

7     " 

-       188,83 

15     " 

131,83 

9      «'          -        318,90 

8     " 

179,97 

16     " 

-       125,97 

1  year,  -     -        300,58 

9     " 

171,91 

17     " 

-       120,31 

2.   "      -     -        271,03 

10     " 

164.46 

18     " 

-       114,89 

3     "      -     -        243,69 

11     " 

157,43 

19     " 

109,70 

4     "      -     -        225,42 

12     " 

-        150,64 

20     " 

-       104,74 

This  table  shows  that  if  8100  be  paid  at  the  birth  of  a  child, 
he  M'ill  be  entitled  to  receive  8376,84,  if  he  lives  to  attain  the 
age  of  21  years.  If  8100  be  paid  when  he  is  ten  years  old,  he 
will  be  entitled  to  receive  8164,46,  if  he  lives  to  attain  the  a^-e 
of  21  years.  And  similarly  for  other  ages.  We  can  easily  find 
by  proportion 

IsL  How  much  must  be  paid,  at  any  age  under  21,  to  pur- 
chase a  given  endowment  at  21  ;  and 

2c?.  What  endowment  a  sum  paid  at  any  age  under  21,  will 
purchase  ? 

EXAMPLES. 

1.  What  endowment,  at  21,  can  be  purchased  for  $250,  paid 
at  the  age  of  10  years  ? 

2.  What  endowment,  at  21,  can  be  purchased  for  8360,  paid 
at  tlie  age  of  5  years  ? 

3.  If  my  child  is  7  years  old.  and  I  purchase  an  endowment 
for  8650,  what  will  he  receive  if  he  attains  the  age  of  21  years? 

4.  If,  at  the  birth  of  a  daugliter,  I  purchase  an  endowment 
for    8350,    what  will  she  receive  if  she  attaiii*  the  ago  of  2] 


years 


*? 


ANNUITIES. 


261 


ANNUITIES. 
258.  An  Annuity  is  a  fixed  sum  of  money  to  be  paid  at 
T'^gular  periods,  either  for  a  limited  time,  or  forever,  in  con- 
rjderalion  of  a  given  sum  paid  in  hand. 

The  Present  Value  of  an  annuity  is  that  sum  which  being 
put  at  compound  interest  would  produce  the  sums  necessary  to 
l);iy  the  annuity. 

The  j)urchaser  of  an  annuity  should  jiay  more  than  tlie  com- 
pound interest  ;  for  the  seller  cannot  aiford  to  take  the  money' 
of  the  purchaser,  invest  it,  re-invest  the  interest,  and  pay  over 
the  entire  proceeds. 

Knowing  the  rate  of  interest  on  money,  and  the  present  value 

of  an  annuity,  a  close  estimate  may  be  made  of  the  price  it 

ought  to  sell  for. 

TABLE 

Shoiving  the  peesent  value  of  an  annuity  of  %l,  from  1  to  30  years,  at 
different  rates  of  iriterct!. 


Years. 

5  per  cent. 

G  per  cent. 

Vear.s. 

5  per  cent. 

G  per  cent. 

-1 

0.95-2381 

0.943396 

16 

10.837770 

10.105895 

1.859410 

1.833393 

17 

11.274066 

10.477260 

3 

2.723X148 

2.673012 

18 

11.689587 

10.827603  i 

4 

3.545950 

3.465106 

19 

12085321 

11.1.58116 

5 

4.329477 

4,212364 

20 

12.462216 

11.469921 

6 

5  075G92 

4.917324 

21 

12,821153 

11.764077 

7 

5.786373 

5.582381 

22 

13.163003 

12-041582 

8 

6.463213 

6.209794 

23 

13488574 

12.303379 

9 

7.107823 

6.801692 

24 

13.798642 

12.550.358 

10 

7.721735 

7.360087 

25 

14.093945 

12  783356 

11 

8.306414 

7.886875 

26 

14.375185 

13.003166 

12 

8.86.3252 

8. .388844 

27 

14  643034 

13.210.5.34 

13 

9393573 

8.852683 

28 

14.898127 

13  406164 

14 

9  898641 

9.2949S4 

29 

15.141074 

13.590721 

15 

10.3796,58 

9.712249 

30 

15  372451 

13.764831 

To  find  the  present  value  of  an  annuity  for  any  rate,  and  for 
any  time,  we  simply  multiply  the  present  value  of  an  annuity 

2.58.  What  is  an  annuity!  What  is  the  present  value  of  an  annuity ! 
How  do  you  find  the  present  value  of  an  annuity  for  a  given  rate  and  time  1 
How  do  you  find  what  annuity  a  given  sum  will  produce,  at  a  given  rate 
and  for  a  given  time  ? 


262  PKRCENTAGE. 

of  SI  for  the  same  rate  and  time,  bj  the  annuity,  and  the  pro- 
duct will  be  its  present  value. 

Thus,  the  present  value  of  an  annuity  of  $600  for  8  years, 
at  G  per  cent,  is 

$6.209794  X  600  =  83725.8764  ;  that  is, 

pres.  val.  of  $1  X  annuity  =  pres.  vaL,  hence, 

pres.  val.  ,,        n 

annuity  = ^ ; ;  therefore, 

pres.  val.  of  §1 

I.  To  find  what  sum  will  produce  a  certain  annuity  at  a  given 
rate,  and  for  a  given  time. 

Multiply  the  present  value  of  an  annuity  of  %1,  at  the  given 
rate  and  for  the  given  time,  hy  the  given  annuity  ;  the  product 
will  he  that  sum. 

II.  To  find  what  annuity  a  given  sum  will  produce  at  a  given 
rate  and  for  a  given  time. 

Divide  the  given  sum,  or  present  value  hy  the  present  value  of 
$1,  for  the  given  rate  and  time,  and  the  quotient  ivill  be  the 
annuity. 

EXAMPLES. 

1.  What  is  the  present  value  of  an  annuity  of  $550,  at  5  per 
cent,  for  21  years? 

2.  What  would  be  the  value  of  an  annuity  that  should  yield 
eight  hundred  and  thirty-five  dollars  a  year  for  sixteen  years, 
the  interest  being  compound,  and  at  the  rate  of  5  per  cent,  per 
annum  ? 

3.  What  is  the  present  value  of  an  annuity  of  $1500  a  year, 
for  30  years,  the  compound  interest  being  reckoned  at  5  per 
cent.  ? 

4.  For  what  sum  could  Mr.  Jones  purchase  an  annuity  for 
twenty-eiglit  years,  of  twelve  hundred  and  twenty  dollars,  the 
compound  interest  being  reckoned  at  6  per  cent.? 

5.  What  annuity,  for  twenty-four  years,  could  be  purchased 
for  tlie  sum  of  twenty-seven  thousand  iivc  luiiulred  and  sixty 
dollars,  the  compound  i-nt crest  being  reckoned  at  6  per  cent.  ? 


ASSESSING    TAXES.  263 

6.  Mr.  Jones  having  a  small  fortune  of  S25000,  and  calcula- 
ting tli;it  he  will  live  about  20  years,  purchases  an  annuity  at 
six  per  cent.,  with  nn  agi-eement  that  he  would  pay  1)20  a  year 
to  an  invalid  sister :  what  was  his  annual  income  from  the 
investment,  after  making  that  payment? 

ASSESSING  TAXES. 

259.  A  Tax  is  a  certain  sum  required  to  be  paid  by  the  in- 
habitants of  a  town,  county,  or  state,  for  the  support  of  govern- 
ment. It  is  generally  collected  from  each  individual,  in  propor- 
tion to  the  amount  of  his  property. 

In  some  states,  however,  every  white  male  citizen  over  the 
age  of  twenty-one  years,  is  required  to  pay  a  certain  tax.  This 
tax  is  called  a  poll-tax ;  and  each  person  so  taxed  is  called  a 
poll. 

2G0.  In  assessing  taxes,  the  first  thing  to  be  done  is  to  make 
a  complete  inventory  of  all  the  property  in  the  town,  on  which 
the  tax  is  to  be  laid.  If  there  is  a  poll-tax,  make  a  full  list  of 
the  polls  and  multiply  the  number  by  the  tax  on  each  poll,  and 
subtract  the  product  from  the  whole  tax  to  be  raised  by  the 
town  ;  the  remainder  will  be  the  amount  to  be  raised  on  the 
properry.  Having  done  this,  divide  the  %ohoh  tax  to  he  raised 
by  the  amount  of  taxable  jjroperty,  and  the  quotient  will  be  the 
tax  on  $1.  Then  multiply  this  quotient  by  the  inventory  of 
each  individual,  and  the  product  will  be  the  tax  on  his  property. 

EXAMPLES. 

1.  A  certain  town  is  to  be  taxed  $4280 ;  the  property  on 
which  the  tax  is  to  be  levied  is  valued  at  $1000000.  Now 
there  are  200  polls,  each  taxed  $1,40.  The  property  of  A  is 
Valued  at  $2800,  and  he  pays  4  polls, 

259.  What  is  tax  ]     How  is  it  generally  collected  1    What  is  a  poll-tax  ? 

2G0.  What  is  the  first  thing  to  be  done  in  assessing  a  tax  1  If  there  is  a 
poll-tax,  how  do  you  find  the  amount  1  How  then  do  you  find  the  per  cent 
of  tax  to  be  levied  on  a  dollar  I  How  do  you  find  the  tax  to  be  raised  on 
each  individual  I 


264 


PERCENTAGE. 


B's  at  $2400,  pays  4  polls, 
C's  at  $2530,  pays  2 
D's  at  $2250,  pays  6 


u 


E's  at  $7242,  pays  4    polls, 
F's  at  $1651,  pays  6       " 
G's  at  $1600,80,  pays  4  « 


What  will  be  the  tax  on  one  dollar,  and  what  will  be  A's  tax, 
and  also,  that  of  each  on  the  list  ? 

First,     $1,40  X  200  =  $280,  amount  of  poll-tax. 
$4280  —  $280  =  $4000,  amount  to  be  levied  on  property. 
Then,     $4000  -^  $1000000  =  4  mills  on  $1. 
Now,  to  find  the  tax  of  each,  as  A's,  for  example, 


A's  inventory, 


$2800 
,004 

11,20 
5,60 


4  polls,  at  $1,40  each,     - 
A's  whole  tax,       _         -         -         $16,80 
In  the  same  manner,  the  tax  of  each  person  in  the  township 
may  be  found. 


ASSESSMENT    TABLE. 

261.  Having  found  the  per  cent,  or  the  amount  to  be  raised 
on  each  dollar,  form  a  table  showing  the  amount  which  certain 
sums  Mould  produce  at  the  same  rate  per  cent.  Thus,  after 
having  found,  as  in  the  last  example,  that  four  mills  are  to  be 
raised  on  every  dollar,  we  can,  by  multiplying  in  succession  by 
the  numbers  1,  2,  3,  4,  5,  6,  7,  8,  &c.,  form  the  following 

TABLE. 


$ 

$ 

$                  $ 

$ 

« 

1 

gives  0,004 

20  gives  0,080 

300  gives   1,200  | 

2 

••      O.OOS 

30      "      0,120 

400 

'       1 .600 

3 

•      0,012 

40      "      0,100 

500 

'       2,000 

4 

•'      0.016 

50      "      0,200 

600 

'       2,t()() 

5 

"       O.O'^O 

60      "      0.210 

700 

'       2,800 

6 

"       0.024 

70      "      0,280 

800 

'       3.200 

7 

"       0.028 

80      "      0,320 

900 

'      3,600 

9 

"      0.0:12 

90      "      0,3()0 

1000 

'      4.000 

<) 

"     0,():!0 

100      "      0,400 

20110 

•      8.000 

10 

"      0.040 

200      "      O.MOO 

3000 

'    12,000 

2G1.   How  do  you  form  tlio  Assessment  table  1 


ASSESSING    TAXES.  265 

This  table  shows  the  amount  to  be  raised  on  each  sum  in  the 
cohimns  under  $'s. 

Note. — If  you  wish  ilie  tax  on  a  sum  not  named  in  the  Table,  as 
$25,  it  is  equal  to  the  sum  of  the  taxes  on  $20  and  So :  and  similarly 
for  other  numbers. 

1.  To  find  the  amount  of  B's  tax  from  this  table. 

B's  tax  on  $2000,  is         -         -         $8,000 
B's  lax  on      400,  is         -         -  1,000 

B's  tax  on  4  polls,  at  $1,40,      -  5, GOO 

B's  total  tax  is       -         -       $15,200 

2.  To  find  the  amount  of  C's  tax  from  the  table. 

C's  tax  on  $2000,  is  -  -  $8,000 

C's  tax  on      500,  is  -  -  2,000 

C's  tax  on        30,  is  -  -  120 

C's  tax  on  2  polls,  is  -  -  2,800 

C's  total  tax  is  -  -  $12,920 

In  a  similar  manner,  Ave  might  find  the  taxes  to  be  paid  by 
D,  E,  &c. 

EXAMPLES. 

1.  In  a  county  embracing  350  polls,  the  amount  of  property 
on  the  tax  list  is  $318200 ;  the  amount  to  be  raised  is  as  fol- 
lows :  for  state  purposes,  $14G5,50  ;  for  county  purposes, 
$350,25;  and  for  town  purposes,  $200,25.  By  a  vote  of  the 
county,  a  tax  is  levied  on  each  poll  of  ^1,50 :  hovi^  much  per 
cent  will  be  laid  upon  the  property  ? 

2.  In  a  county  embracing  a  population  of  98415  persons,  a 
tax  is  levied  for  town,  county,  and  state  purposes,  amounting  to 
S100406.  Of  this  sum,  a  part  is  to  be  raised  by  a  tax  of  25 
cents  on  eacli  poll,  and  the  remainder  by  a  tax  of  two  mills  on 
the  dollar  :  what  was  the  amount  of  property  on  the  tax  list  ? 

3.  In  a  county  embracing  a  popuhuion  of  5G450  persons,  a 
tax  is  levied  for  town,  county,  and  state  purposes,  amounting  to 
887467  :  the  pergonal  and  real  estate  is  valued  at  S4890300. 
Each  poll  is  taxed  25  cents  :  what  per  cent  is  the  tax,  and  how 


266  PEKOENTAGE. 

much  will  a  man's  tax  be,  who  pays  foi-  5  polls,  and  whose  px-o- 
perty  is  valued  at.  $5400  ? 

Yy''hat  is  B's  lux,  who  was  assessed  for  2  polls,  and  whose 
j)rpperty  was  valued  at  ^3760,50  ? 

4.  A  banking  corporation,  consisting  of  40  persons,  was  taxed 
$957.p0  ;  their  property  was  valued  at  §125000,  and  each  poll 
was  assessed  50  cents  each  :  what  per  cent  was  their  tax,  and 
what  was  a  man's  tax,  who  paid  for  1  poll,  and  whose  share 
was  assessed  for  $2000  ? 

5.  What  sum  must  be  assessed  to  raise  a  net  amount  of 
$5674,50,  allowing  2^  per  cent  commission  on  the  money  col- 
lected  (Art.  218)  ? 

6.  Allowing  4  per  cent,  for  collection,  what  sum  must  be 
assessed  to  raise  $21346,75  net  ? 

7.  In  a  certain  township,  it  becomes  necessary  to  levy  a  tax  of 
14423,2475,  to  build  a  public  hall.  The  taxable  property  is 
valued  at  $916210.  and  the  town  contains  150  polls,  which  are 
each  assessed  50  cents.  "What  amount  of  tax  must  be  raised 
to  build  the  hall,  and  pay  5  per  cent,  for  collection,  and  what  is 
the  tax  on  a  dollar  ? 

What  is  a  person's  tax  who  pays  for  3  polls,  and  whose  per- 
sonal property  is  valued  at  $2100,  and  his  real  estate  at  $3000  ? 
What  is  G  s  tax,  who  is  assessed  for  1  poll,  and  $1275,50  ? 
What  is  li's  tax,  who  is  assessed  for  1  i)oll,  and  $2456  ? 

8.  The  people  of  a  school  district  wish  to  build  a  new  schoo. 
house,  which  shall  co-^^t  ^2850.  The  taxable  ])roperty  of  tin- 
district  is  valued  at  $190000  :  Avhat  will  be  the  tax  on  a  dollar, 
and  what  will  be  a  man's  tax,  whose  property  is  valued  at  $7500  ? 

How  much  is  IMr.  Merchant's  tax,  whose  personal  and  real 
estate  are  assessed  for  $1200  ? 

9.  In  a  school  district,  a  school  is  supported  by  a  rate-bill. 
A  teacher  is  employed  for  6  months,  at  $60  a  month  ;  the  fuel 
and  other  contingoucies  amount  to  $66.  They  drew  $41,60 
public  money,  and  the  whole  number  of  days  attendance  was 
7688:  what  was  D's  tax,  who  sent  148  days? 

AVhat  was  F's  tax,  wlio  sent  184.V  days? 


CUSTOM    HOUSE   BUSINESS.  2G7 

CUSTOM  HOUSE  BUSINESS. 

259.  Persons  who  bring  goods,  or  merchandise,  into  the 
United  States,  from  foreign  countries,  are  required  to  land 
them  at  particular  places  or  ports,  called  Ports  of  Entry,  and 
to  pay  a  certain  amount  on  their  value,  called  a  Duty.  This 
duty  is  imposed  by  the  General  Government,  and  must  be  the 
game  on  the  same  articles  of  merchandise,  in  every  part  of  the 
United  States. 

Besides  the  duties  on  merchandise,  vessels  employed  in  com- 
merce are  required,  by  law,  to  pay  certain  sums  for  the  privilege 
of  entering  the  ports.  These  sums  are  large  or  small,  in  pro- 
portion to  the  size  or  tonnage  of  vessels.  The  moneys  arising 
from  duties  and  tonnage,  are  called  revenues. 

260.  The  revenues  of  the  country  are  under  the  general 
direction  of  the  Secretary  of  the  Treasury,  and  to  secure  tlieir 
faithful  collection,  the  government  has  appointed  various  othcera 
at  each  port  of  entry  or  place  where  goods  may  be  landed. 

261.  The  office  established  by  the  government  at  any  port 
of  entry,  is  called  a  Custom  House,  and  the  officers  attached  to 
it  are  called  Custom  House  Officers. 

262.  All  duties  levied  by  law  on  goods  imported  into  the 
United  States,  are  collected  at  the  various  custom  houses,  and 
are  of  two  kin,ds — Specific  and  Ad  valorem. 

Specific  Duty  is  a  certain  sum  on  a  particular  kind  of 
goods  named  ;  as  so  much  per  square  yard  on  cotton  or  woollen 
cloths,  so  much  per  ton  weight  on  iron,  or  so  much  per  gallon 
on  molasses. 


259.  What  is  a  port  of  entry  ?  What  is  a  duty  ?  By  wliom  are  duties 
imposed  ?  What  charges  are  vessels  required  to  pay  !  What  arc  the 
moneys  arising  from  duties  and  tonnage  called  1 

260.  Under  whose  direction  are  the  revenues  of  the  country  1 

261.  What  is  a  custom  house  \  What  are  the  officers  attached  to  it  called  1 

262.  Where  are  the  duties  collected  !  How  many  kinds  are  there,  and 
what  are  they  called  1     What  is  a  specific  dutyl     An  ad  valorem  dutyl 


268  PERCENTAGE. 

Ad  valorem  Duty  is  such  a  per  cent  on  the  actual  cost  of 
the  goods  in  the  country  from  which  they  are  imported.  Thus, 
an  ad  valoi-em  duty  of  15  per  cent  on  Enghsh  cloths,  is  a  duty 
of  15  per  cent  on  the  cost  of  cloths  imported  from  England. 

263.  The  laws  of  Congress  provide,  that  the  cargoes  of  all 
TC-ssels  freighted  with  foreign  goods  or  merchandise,  shall  be 
weighed  or  gauged  by  the  custom  house  officers  at  the  port  to 
■vs'hich  they  are  consigned.  As  duties  are  only  to  be  paid  on  the 
articles,  and  not  on  the  boxes,  casks,  and  bags  which  contain 
them,  certain  deductions  are  made  from  the  weights  and  mea- 
sures, called  Allowances. 

Gross  Weight  is  the  whole  weight  of  the  goods,  together  with 
that  of  the  hogshead,  barrel,  box,  bag,  &c.,  wliich  contains  them. 

Net  Weight  is  what  remains  after  all  deductions  are  made. 

Draft  is  an  allowance  from  the  gross  weight  on  account  of 
waste,  where  thei'e  is  not  actual  tare. 

lb.  lb. 

On         112  it  is  1, 

From       112  to     224        "     2, 

"  224  to     336        "     3, 

«  33G  to  1120        "     4, 

«        1120  to  2016        ''     7, 

Above   2016  any  weight   "     9, 

consequently,  9//;.  is  the  greatest  draft  generally  allowed. 

Tare  is  an  allowance  made  for  the  weight  of  the  boxes,  bar- 
rels, or  bags  containing  the  commodity,  and  is  of  three  kinds, 
1st.  Legal  tare,  or  such  as  is  established  by  law;  2d.  Cus- 
tomary tare,  or  such  as  is  established  by  the  custom  among 
merchants ;  and  3d.  Actual   tare,  or  such  as  is  found  by  re- 


203.  Wliat  do  tlie  laws  of  Coniiress  direct  in  relation  to  forciirn  rroods  1 
Why  arc  deductions  made  from  their  weight  ?  Wiiat  are  these  dcductioiia 
called?  Wiiat  is  gross  \vci;i;ht  ?  What  is  net  weight !  "\\'hat  is  draft ' 
What  is  the  greatest  draft  allowed  !  What  is  tare?  What  arc  the  dill'er 
eiit  kinds  of  tare  1     AVUat  allowances  are  made  on  lic^uors  1 


CUSTOM    HOUSE    BUSINESS.  269 

moving  the  goods  and  actually  wcigliing  the  boxes  or  casks  in 
which  they  are  contained. 

On  liquors  in  casks,  customary  tare  is  sometimes  allowed  on 
the  supposition  that  the  cask  is  not  full,  or  what  is  called  its 
actual  ivants  ;  and  then  an  allowance  of  5  per  cent  for  leakage. 

A  tare  of  10  per  cent  is  allowed  on  porter,  ale,  and  beer,  in 
bottles,  on  account  of  breakage,  and  5  per  cent  on  all  other 
liquors  in  bottles.  At  the  custom  house,  bottles  of  the  common 
size  are  estimated  to  contain  2|-  gallons  the  dozen.  For  tables 
of  Tare  and  Duty,  see  Ogden  on  the  Tarift'of  1842. 

EXAMPLES. 

1.  What  is  the  net  weight  of  25  hogsheads  of  sugar,  the 
gross  weight  being  66c«'i.  ^qr.  lUb. ;  tare  lllb.  per  hogshead? 

civt,  qr.  lb. 

66  3  14  gross. 

25  X  11  =  275/6.     -     -     2  3  tare. 

Ans.M  0  14  net. 


2.  If  the  tare  be  Alb.  per  hundred,  what  will  be  the  tare  en 
QT.  2cwt.  3qr.  Ulb.  ? 

Tare  for  QT.  or  UOcwf.  =  ASOlb. 
2cwt.  z=      8 
Sqr.    =       3 
Ulb.     =       0^ 

Tare     -     -     jM^Ib. 

3.  What  will  be  the  cost  of  3  hogsheads  of  tobacco  at  $9,47 
per  cwt.  net,  the'gross  weight  and  tare  being  of 

ctvt.  qr.  lb.  lb. 

No.  1     -     -       9     3  24  -  -  tare  146 

«     2     -     -     10     2  12  -  -  "  150 

"     3     -    -     11     1  24  -  -  «  158 

4.  At  21  cents  per  lb.,  what  will  be  the  cost  of  5hkd.  of  coffee, 
the  tare  and  gross  weight  being  as  follows  : 


570 


1'! 

1  i£Ci 

■;]SiTAGK. 

cwl. 

qr. 

/<;-. 

lb. 

No.  1     ■ 

■     -     6 

2 

14 

-     - 

tare    94 

«     2     - 

■     -     9 

1 

20 

-     - 

"     100 

"     3     ■ 

■     -     G 

2 

22 

-     - 

"       88 

"    >     ■ 

•     -     7 

2 

24 

-     - 

"       89 

"    *5     - 

•     -     8 

0 

13 

- 

"     100 

5.  What  is  the  net  weight  of  IShhcL  of  tobacco,  each  weigh- 
ing gross  Sctvf.  dqr.  1Mb.',  tare  16/6.  to  the  cwt."? 

G.  In  47'.  Zciol.  Zqr.  gross,  tare  20/i.  to  the  cwt.,  Avhat  is  the 
net  weight  ? 

7.  What  is  the  net  weight  and  value  of  80  kegs  of  figs, 
gross  weight  IT.  Wcict.  3q}:,  tax*e  12/6.  per  cwt.,  at  $2,31 
per  civi. 

8.  A  merchant  bought  Idcwt.  Iqr.  24/6.  gross  of  tobacco  in 
leaf,  at  $24,28  per  cwl.;  and  12ca'L  3qr.  19/6.  gross  in  rolls,  at 
$28,56  per  cwt. ;  the  tare  of  the  former  Avas  149/6.,  and  of  the 
latter    49/6. :  what  did  the  tobacco  cost  him  net  ? 

9.  A  grocer  bought  17^^h/id.  of  sugar,  each  lOart.  Iqr.  14/6., 
draft  7/6.  per  civ/.,  tare  4/6.  per  cwt.  What  is  the  value  at 
$7,50  per  art.  net? 

10.  In  29  parcels,  each  weighing  Sctvt.  oqr.  14/6.  gross,  draft 
8/6.  per  cwt.,  tare  4:'6.  per  cwt.  how  much  net  weiglit,  and 
what  is  the  value  at  $7,50  per  cwt.  net  ? 

11.  A  merchant  bought  7  hogsheads  of  molasses,  each  Aveigh- 
ing  4rwf.  'dqr.  HI/),  gross,  draft  7/6.  per  cwt.,  tare  8/6.  per 
hogshead,  and  damage  in  the  whole  99|-/6. :  what  is  the  value 
at  $8,45  per  cwt.  net  ? 

12.  Tlie  net  value  of  a  hogshead  of  Barbadoos  sugar  was 
$22,50;  the  custom  and  fees  $12,49,  freight  $5,11,  factorage 
$1,31  ;  the  gross  weight  was  llcwf.  Iqr.  15/6.,  tare  ll]-/6.  per 
cirf. :  what  was  the  sugar  rated  at  per  cwt.  net,  in  the  bill  of 
parcels  ? 

13.  In  7h/id.  of  .sugar,  each  weighing  3cwt.  2qr.  14/6.  gross, 
tare  21/6.  per  cwt.,  what  is  the  value  at  $6,25  per  ca-t.  ? 


CUST<m    UOUSE   BUSINESS.  271 

14.  I  have  imported  87  jars  of  Lucca  oil,  each  containing 
47  gallons  :  Avhat  did  the  freight  come  to  at  $1,19  per  act.  net, 
reckoning  \lh.  in  Wlb.  for  tare,  and  9//^  of  oil  to  the  gallon  ? 

15.  A  grocer  bought  bhhd.  of  sugar,  each  weighing  V^cwt. 
\qr.  12//j.,  at  7-^  cents  a  pound  ;  the  draft  was  l^lh.  per  civi.^ 
and  the  tare  5^  per  cent :  wliat  was  the  cost  of  the  net  weight  ? 

]  6.  A  wholesale  merchant  receives  450  bags  of  coffee,  each 
Aveighing  76  pounds;  the  tare  was  8  per  cent,  and  the  invoice 
price  lOi  cents  per  pound.  He  sold  it  at  an  advance  of  33^ 
per  cent :  what  was  his  whole  gain,  and  what  his  selling  price  ? 

17.  A  merchant  imported  17G  pieces  of  broadcloth,  each 
piece  measuring  46iyt/.,  at  $3,25  a  yard  :  what  will  be  the  duty 
at  30  per  cent  ? 

18.  What  is  the  duty  on  547^.  13c',y^.  Zqr.  2<dlb.  of  iron,  in- 
voiced at  $45  a  ton,  and  the  duty  33^  per  cent  ? 

19.  What  is  the  ad  valorem  duty,  at  25  per  cent,  on  ZhJid. 
of  molasses,  at  35  cents  a  gallon,  an  allowance  of  2  per  cent 
beinii:  made  for  leakage  ? 

20.  If  I  import  50  chests  of  tea,  each  weighing  140  pounds, 
invoiced  at  60  cents  a  pound,  a  deduction  of  IQlh.  per  cwt.  being 
made  for  tare  :  what  were  the  governmental  duties,  at  40  per 
cent  ad  valorem  ? 

21.  What  will  be  the  duty  on  225  bags  of  coffee,  each  weigh- 
ing gross  160/6.,  invoiced  at  6  cents  per  lb.;  2  per  cent  being 
the  legal  rate  of  tare,  and  20  per  cent  the  duty? 

22.  What  duty  must  be  paid  on  275  dozen  bottles  of  claret, 
estimated  to  contain  2|  gallons  per  dozen,  5  per  cent  being 
allowed  for  breakage,  and  the  duty  being  35  cents  per  gallon  ? 

23.  A  merchant  imports  175  cases  of  indigo,  each  case  weigh- 
ing 196/6.  gross  :  15  per  cent  is  the  customary  rate  of  tare,  and 
the  duty  5  cents  per  lb.     W^hat  duty  must  he  pay  on  the  whole  ? 

24.  What  is  the  tare  and  duty  on  75  casks  of  Epsom  salts, 
each  weigiiing  gross  2cwt.  ^qr.  24//;.,  and  invoiced  at  1|-  cents 
per  lb.,  the  customary  tare  being  11  per  cent,  and  the  rate  of 
duty  20  per  cent  ? 


272  EQTJATION    OF    PAYMENTS.  [ 

EQUATION  OF  PAYMENTS.  i 

264.  Equation  op  Payments  is  a  process  of  finding  the    }| 
average  time  of  payment  of  several  sums  due  at  different  times, 
so  that  no  interest  shall  be  gained  or  lost.* 


OPERATION. 

$15 

X 

6 

=  90 

$18 

X 

7 

==  126 

$24 

X 

10 

=  240 

57 

5 

7)456(8 
456 

1.  B  owes  Mr.  Jones  $57:  $15  is  to  be  paid  in  6  months  ; 
$18  in  7  months  ;  and  $24  in  10  months  :  what  is  the  average 
time  of  payment  so  that  no  interest  shall  be  gained  or  lost  ? 

Analysis. — The  interest  of  $15  for  6 
months,  is  the  same  as  the  interest  of 
$1  for  90  months:  the  interest  of  $18 
for  7  mouths  is  the  same  as  the  interest 
of  $1  for  126  months;  and  the  interest 
of  $24  for  10  months  is  the  same  as  the 
interest  of  $1  for  240  months;  hence, 

the  sum  of  these  products,  466,  is  the  number  of  months  it  would 
take  $1  to  produce  the  required  interests.  Now^,  the  sum  of  the 
payments,  $57,  will  produce  the  same  interest  in  one  fifty-seventh 
part  of  the  time  ;  that  is,  in  8  months  :  hence,  to  find  the  average 
time  of  payment : 

Multiply  each  "payment  by  the  time  before  it  becomes  due,  and 
divide  the  sum  of  the  products  by  the  sum  of  the  payments  :  the 
quotient  ivill  be  the  average  time. 

EXAMPLES. 

1.  A  merchant  owes  Si 200,  of  which  $200  is  to  be  paid  id 
4  months,  $400  in  10  months,  and  the  remainder  in  16  months; 
if  he  pays  the  Avliole  at  once,  at  what  time  must  he  make  the 
payment  ? 

264.  What  is  equation  of  payments  1  How  do  you  find  the  average 
fime,  of  payment  \ 

*  The  mean  time  of  payment  is  sometimes  found  by  first  fiiulinjr  tht 
prr.scnL  value  of  each  payment  ;  but  the  rule  here  <rivcn  has  the  sanction 
of  the  best  authorities  in  this  country  and  FJngland. 


•n 


EQUATION    OF    J'Ai'MENTS.  273 

2.  A  owes  B  $2400 ;  one-third  is  to  be  paid  in  6  months, 
one-fourth  in  8  months,  and  the  remainder  in  12  months  :  what 
is  the  mean  time  of  payment  ? 

3.  A  merchant  has  due  him  $4500 ;  one-sixth  is  to  be  paid 
in  4  months,  one-tliird  in  6  months,  and  the  rest  in  12  mouths: 
what  is  the  equated  time  for  the  payment  of  the  whole  ? 

4.  A  owes  B  $1200,  of  which  $240  is  to  be  paid  in  three 
months,  $360  in  five  months,  and  the  remainder  in  ten  months : 
what  is  the  average  time  of  payment  ? 

5.  Mr.  Swain  bought  goods  to  the  amount  of  $3840,  to  t)e 
paid  for  as  follows,  viz. :  one-fourth  in  cash,  one-fourth  in  6 
months,  one-fourth  in  7  months,  and  the  I'emainder  in  one  year : 
what  is  the  average  time  of  payment  ? 

6.  A  flour  merchant  bought  at  one  time  150  barrels  of  flour, 
at  $8  a  barrel ;  15  days  afterwards  he  bought  176  barrels,  at 
$8,50  a  barrel ;  25  days  after  that  he  bought  200  barrels,  at 
$9  a  barrel :  how  many  days  after  the  first  purchase  would  be 
the  equated  time  of  payment  ? 

7.  A  man  bought  a  farm  for  $5000,  for  which  he  agreed  to 
pay  $1000  down,  $1200  in  3  months,  $800  in  8  months,  $1500 
in  10  months,  and  the  remainder  in  one  year:  if  he  pays  the 
whole  at  once,  what  would  be  the  average  time  of  payment  ? 

Notes. — 1.  In  finding  the  equated  time  of  payments  for  several 
Bums,  due  at  different  times,  any  day  may  be  assumed  as  the  one  from 
which  we  reckon. 

2.  If  one  of  the  payments  is  due  on  the  day  from  which  the 
equated  time  is  reckoned,  its  corresponding  product  will  be  nothing,  but 
the  payment  must  still  be  added  in  finding  the  sum  of  the  payments. 

8.  A  person  owes  three  notes  :  the  first  is  for  $200,  payable 
July  1st ;  the  second  for  $150,  payable  August  1st ;  and  the 
third  for  $250,  payable  August  15th  :  what  is  the  average 
time,  reckoned  from  July  1st  ? 

Notes  — 1.  May  the  equated  time  be  reckoned  from  any  dayt 
2.  Tf  one  of  the  payments  is  due  on  the  day  from  which  the  equated 
time  is  reckoned,  what  will  be  the  value  of  the  corresponding  product  1 


27J:  EQUATION    OI    PAY3IE^-TS. 

9.  E.  BoxD,  Bought  of  Trust  &  Co. 
1856,  Aug.   1,  450  yds.  muslin,  at  10  cents,  -    -    845,00 

«       "      16,  800    "     caljco     "    12i   "  -  -    100.00 

"     Sept.   5,  720    «  bombazine    80      -  -  -    576,00 

"     Oct.     1,  300    "     cloth,  at  3,50      -  -  -  1050,00 
On  what  day  does  the  whole  amount  fall  due  ? 

10.  Mr.  Johnson  sold,  on  a  credit  of  8  months,  the  following 
bills  of  goods : 

April  1st,  a  bill  of  $4350, 
May  7th,  a  bill  of    3750, 
June  5th,  a  bill  of    2550. 
At  what  time  will  the  whole  become  due  ? 

11.  A  purchased  of  B  the  following  bill  of  goods,  on  differ- 
ent times  of  credit : 

May  1st,       1857,  a  bill  amounting  to     $800  on  3  months. 

June  1st,         "  "  "  "        700    "  3       " 

«    25th,      "  "  "  "        900    "  4       " 

July  25th,      "  "  "  "      1000    "  6       " 

"What  is  tlie  equated  time  for  the  payment  of  the  whole,  and  on 
what  day,  reckoned  from  Aug.  1st,  is  the  bill  due  ? 

12.  A  person  purchased  the  following  bills  of  goods,  on  dif- 
ferent times  of  credit : 

Jan.  1st,       1855,  a  bill  amounting  to  $367,20  on  4  months. 

"  "  "     901,80   "  3       " 

"  "  "     826,38   "  5       " 

«  «  "     854,88    "  6       " 

«  "  «     396,50    "  4       " 

What  is  the   average  time  of  payment  from  the   time  the 
first  bill  falls  due  ? 

264.*    To  find  liow  long  a  sum  of  money  must  he  at  interest^ 
to  balance  the  interest  on  a  given  sum  for  a  given  time. 

1 .  If  A  lends  B    $700  for  3  months,  how  long  ought  B  to 
lend  A  $500  to  balance  tJiu  interest  ? 


"     28th, 

(( 

Feb.  24th, 

(( 

March  30th, 

ii 

May  1st, 

ii 

EQUATION   OF   PAYMENTS.  275* 

Analysis. — Since  $700,  in  3  months,  will  pro.        operat-.on. 

duce  as   much   interest  as  S2100  in   1  month,  it  '-qo  x  3  =  2100- 

will  require    as  many  months   for   $500    to   pro-  2100 — 500 --44 
duce  the  same  interest,  as  500  is  contained  times 
in  2100  :  \yhich  is  4-^. 

2.  A  lends  B  his  note  for  $900  payable  iu  5  months  :  how 
long  should  B  lend  A  his  note  for  $480,  to  balance  the  favor  ? 

Ans.   9~  months. 

3.  C  buys  of  D,  100  barrels  of  flour,  at  87^-  per  barrel,  and 
in  payment  gives  his  note  for  3  months  ;  D  buys  of  C,  500 
bushels  of  Avheat  at  80  cents  per  bushel,  and  gives  his  note  in 
payment :  how  long  must  this  note  run,  that  each  may  have  an 
equal  use  of  the  other's  money  ?  A?is.  o|  months. 

To  find  hoio  long  the  balance  may  he  kept  when  'paynitnts  are 
made  before  they  are  due. 

1.  A  owes  B,  6'800  payable  in  G  months  ;  at  the  expiration 
of  4  months,  he  pays  $500  :  how  long  beyond  the  6  months 
should  A  retain  the  balance,  so  that  neither  shall  make  or  lose 
interest  ? 

Analysis. — A  has  the   right   to  retain  the  $800 
for  6  months,  or  $4800   for  1   month.     He    retains      operation. 
$500  for  4  months,  or  $2000  for   1  month.     Hence,    800x6  =  4800 
he  may  still   retain   S2800  for  1  month,  or  the  bal-    500X4  =  2000 
ance,  $300,  as   many  months   as   300   is   contained    300  2800 

times  in  2800  ;  or,  9|-  months  from  the  date  of  the    2800-f-300  =  9^ 
debt ;  or  9^—6  =  3^  months  beyond  the  time  of  six 
months. 

2.  C  owes  D,  $2500  payable  in  4  months,  but  at  the  end  of 
3  months  pays  him  $1600  :  how  long  after  the  payment  of 
$1600  should  the  remainder  be  retained  to  balance  the  ac- 
count? Ans.  2^  months. 

3.  One  merchant  owes  another  $1600  payable  in  6  months, 
but  at  the  end  of  3  months  pays  $400  ;  at  the  end  of  4  months, 
$400,  and  at  the  end  of  5  months  $300  ;  how  long,  from  the 
last  payment,  may  the  balance  be  retained  to  square  the  ac- 
count ?  A71S.   51  months. 


27G*  EQUATION    OF    PAYMENTS. 

4.  A  note  for  $500  dated  November  6th,  1856,  payable  in 
3  months,  was  given  by  E  to  F.  On  December  3d,  E  paid 
$350  :  when  ought  the  remainder  to  be  paid  to  balance  the 
account?  .4ns.  July  3rd,  1857. 

264.**  An  account  is  said  to  be  balanced  when  the  sum  of 
the  items  on  the  debtor  side  is  equal  to  the  sum  of  the  items  on 
the  credit  side.  When  these  two  sums  are  unequal,  such  an 
amount  is  added  to  the  less  as  will  make  the  sum  equal  to  the 
<Treater.  This  is  called  the  balance.  There  are  three  kinds 
of  balances  : 

1st.  The  merchandize  balance,  in  which  interest  on  the  items 
is  not  considered. 

2d.  The  interest  balance,  which  adjusts  the  interest  on  the 
two  sides  of  an  account ;  and 

3d.  The  cash  balance,  which  arises  from  combining  the 
merchandize  balance  with  the  interest  balance. 

Accounts  are  settled  either  by  cash  or  by  note.  In  ascer- 
taining the  cash  balance  of  an  account,  interest  is  allowed  on 
all  the  items  of  both  sides  ;  the  balance  of  interest  makes  a 
new  item,  and  may  belong  to  either  side  of  the  account. 

1.  Ascertain  the  cash  balance  of  the  following  account  on 
the  25th  of  April,  1850. 

Dr.  S.  Snodgrass,  Cr. 


1850.   April  Is^t,  To  goods,  $375.00 

"    I7th,   "        "        268,00 

'=    25th,    ''        "         175,00 

Cash  balance,  237.93 

$1055,93 


April     7tli,  By  goods,   $675,00 
"     15th,     "      "  380,0t) 

'•     25th,  Bal.  of  Int. ,93 

$1055^93 


Analysis. — Reckoning  backwards  from  April  25tli,  we  find  the 
days  for  which  we  charge  interest,  and  these  are  used  as  multipliers. 
The   interest  of  $375  for  24   days   is  the  same  as  the  interest  of  $1 


EQUATION   OP   PAYMENTS.  277* 

for  9000  days;  and  so  of  the  other  items.  The  difference  of  tho 
sums  of  these  products,  is  the  number  of  days  which  $1  must  be  at 
interest  to  produce  the  baUincc  of  interest,  and  the  balance  always 
goes  with  the  larger  sum  of  the  products. 


OPERATION. 

Debtor  Items.                                         Creditor  Items 

375  X  24  =  9000 

675  X  18  =  12150 

268  X     8  =  2144 

380  X  10  =    3800 

175  X    0  =  0000 

15950 

11144 

11144 

4806 
Then,  4806  "^tVo  =  ^-^3  +,  balance  of  interest. 

Rule. — I.    Take  the  latest   date  of  the  account,  or  any  later 
date  at  which  the  balance  is  to  be  struck,  as  the  point   of  rerJc- 
oning,   and  find  the    days  between   this  date  and  the  date   of 
each  item  ;  and  consider  these  days  as  multipliers. 

II.  Multiply  each  item  by  its  midtiplier  ;  then  take  the  dljjer- 
ence  of  the  sums  of  these  2'>'>^odiicts,  and  multiply  it  by  the  inter- 
est for  1  day :  the  result  will  be  the  interest  balance,  which  is  to 
he  added  to  the  side  having  the  greater  sum. 

III.  Then  find  the  cash  balance. 

Notes. — 1 .  If  the  cash  balance  had  been  required  on  any  day  after 
the  25th  of  April,  the  mode  of  proceeding  would  have  been  exactly 
the  same. 

2.  In  the  examples  the  rate  of  interest  will  be  taken  at  7  per 
cent,  and  360  days  in  the  year. 

3.  After  the  balance  of  interest  is  found,  the  cash  balance  is  ob- 
tained, by  adding  the  two  sides  of  the  account,  taking  the  difference 
of  the  sums  and  placing  it  on  the  s/naller  side  of  the  account. 

4.  If  the  cash  balance  is  settled  by  a  note,  interest  shoukl  run  ci 
the  note  from  the  date  of  the  cash  balance  to  the  time  of  payment. 

5.  Let  the  pupil  find  the  interest  and  cash  balance  in  each  of  th" 
following  examples. 


278' 


EQUATION'   OF   PAYMENTS. 


2.  What  is  the  balance  of  interest  and  what  the  cash  balance 
on  the  following  account  on  Marcli  20th  ? 


Dr. 

S.   JOHXSON. 

Or. 

856.   Jan.  1,  To  merch. 

$500      ' 

Jan.  5,     By  cash, 

$350 

■•  16,    ''    cash, 

450 

•'  19,      "    merch. 

780 

Feb.  5,    "    merch. 

680 

"  25,      '•        " 

250 

>•  24,    "        " 

300 

Feb.  15,    "    cash, 

600 

Mar.     1,  "    cash, 

150 

cash  balance, 

700,65 

"    16,  •'•'    merch. 

600 

$2680,65 

Interest  balance, 

,65 
12680,65 

Note. — 1.  When  the  items  have  the  same  or  different  times  of 
credit  allowed,  find  when,  the  items  are  payable  and  then  proceed  as 
before. 

2.  If  the  cash  balance  is  required  on  a  day  previous  to  the  latest 
date  of  the  items,  find  the  ca.sh  balance  for  this  latest  date  :  then  find 
the  present  value  for  the  given  date :  this  will  be  the  cash  balance. 

3.  Allowing  a  credit  of  six  months  on  each  item,  what  is  the 
interest  and  cash  balance  Feb.  1st,  1856  ? 

R.  Sherman.  Cr. 

Feb'y  6,  By  merch.  $800 
i\Iar.    7,     "         "       900 
interest  bal.  45,48 

$2150  cash  bal.  404,52 

2150.00 

4.  Allowing  a  credit  of  3  months  on  each  of  the  items  of 
the  following  account,  what  would  be  the  interest  and  cash 
balance  on  October  31st,  1856. 

Dr.  R.  Rivers.  '  Or. 


Dr. 

1855.   July  1st,  To  merch.,  $750 

"    17th,  "         "          600 

"   25lh,  "         "          800 


1856.  May  1,   To  merch.  $500 

"    20,  "  675 

Jun.  6,     To  cash,     350 

July  9;     merch.        175 

cash  balance,  620  ,70 

$232o7n) 


May      6th,  By  cash,  $400 


25th. 


mer. 


620 
900 
July    20th,    '■'■  mer.        400 
interest  balance,  .70 


June   16th,    "  cash, 

c 


ALLIGATION.  275 

ALLIGATION. 

265.  Alligation  is  that  branch  of  Arithmetic  which  treats 
of  all  questions  relating  to  the  mixing  or  compounding  of  two 
or  more  ingredients  of  different  values.  It  is  divided  into  two 
parts  :  Alligation  Medial  and  Alligation  Alternate. 

ALLIGATION   MEDIAL. 

266.  Alligation  Medial  teaches  the  method  of  findins; 
the  price  or  quality  of  a  mixture  of  several  simple  ingredients 
whose  prices  or   qualities   are  known. 

1.  A  grocer  would  mix  200  pounds  of  lump  sugar,  Avorth  13 
cents  a  pound,  400  pounds  of  Havana,  worth  10  cents  a  pound, 
and  GOO  New  Orleans,  worth  7  cents  a  pound  ;  what  should  be 
the  price  of  the  mixture  ? 

OPERATION. 

Analysis. — The  quantity,  200/6.,  200  x  13  =  26.00 

at  13  cents  a  pound,  cost.s  $26 ;  400  400  x  10  =  40.00 

pounds,  at   10  cents  a  pound,  costs  600  x     7  =  42.00 

$40  ;  and  600/6.  at  7  cents  a  pound,  1200  )  108,00(9  cts. 

costs   $42 :    hence,  the  entire  mix- 
ture, consisting  of  1200/6.,  costs  $108.    Now,  the  price  of  the  mixture 
will  be   as  many  cents  as  1200  is  contained  times  in  10800  cents : 
viz.,  9  times.     Hence,  to  find  the  price  of  the  mixture. 

Rule. — Multipli/  the  inice  or  quality  of  a  unit  of  each  simple 
hy  the  number  of  such  units  :  take  the  sum  of  their  products 
and  divide  it  by  the  ivhole  number  of  units  :  the  quotient  will  be 
the  price  or  quality  of  a  miit  of  the  mixture. 

EXAMPLES. 

1.  If  1  gallon  of  molasses,  at  75  cents,  and  3  gallons,  at  50 
cents,  be  mixed  with  2  gallons,  at  37|^,  what  is  the  mixture 
worth  a  gallon  ? 

265.  What  is  Alligation  1  Into  how  many  parts  is  it  divided  1  What 
are  they  1 

266  What  is  Alligation  Medial  \  IIow  do  you  find  the  prict-  of  the 
mixture  1 


276  ALLIGATION. 

2.  If  teas  at  371,  50,  62i,  80,  and  100  cents  per  pound,  be 
mixed  together,  what  ay  ill  be  the  value  of  a  pound  of  the 
mixture  ? 

3.  If  5  gallons  of  alcohol,  worth  GO  cents  a  gallon,  and  3 
gallons  worth  96  cents  a  gallon,  be  diluted  by  4  gallons  of  water, 
what  will  be  the  price  of  one  gallon  of  the  mixture  ? 

4.  A  farmer  sold  50  bushels  of  wheat  at  $2  a  bushel ;  GO 
bushels  of  rye,  at  90  cents;  36  bushels  of  corn,  at  G2i  cents; 
and  50  bushels  of  oats,  at  39  cents  a  bushel :  what  was  the 
average  price  per  bushel  of  the  whole  ? 

5.  During  the  seven  days  of  the  week,  the  thermometer 
stood  as  follows  :  70°,  73°,  731°,  77°,  70°,  801°,  and  81°  :  what 
was  the  average  temperature  for  the  w^eek  ? 

6.  If  gold  18,  21,  17,  19,  and  20  carats  fine,  be  melted  to- 
gether, what  will  be  the  fineness  of  the  compound  ? 

7.  A  grocer  bought  3Alb.  of  sugar  at  5  cents  a  pound,  1021b. 
at  8  cents,  lS6lb.  at  10  cents  a  pound,  and  34/6.  at  12  cents  a 
pound.  He  mixed  it  together,  and  sold  the  mixture  so  as  to 
make  50  per  cent  on  the  cost:  what  did  he  sell  it  for  per 
pound  ? 

8.  A  merchant  sold  8//;.  of  tea,  11^5.  of  coifee,  and  25lb.  of 
sugar,  at  an  average  of  15  cents  a  pound.  The  tea  was  worth 
30  cents  a  pound ;  the  coffee,  25  cents  a  pound ;  and  the  sugar, 
7  cents  a  pound  :  did  he  gain  or  lose,  and  how  much  ? 


ALLIGATION   ALTERNATE. 

267.  Alligation  Alternate  teaches  the  method  of  finding 
wliat  proportion  of  several  simples,  whose  prices  or  qualities 
are  known,  must  be  taken  to  form  a  mixture  of  any  required 
price  or  quality.  It  is  the  reverse  of  Alligation  Medial,  and 
may  be  proved  by  it. 


267.  What  is  Alligation  Alternate  1     How  may  Alligation  Alternate  b« 
proved ! 

2G8.  How  do  you  find  the  proportional  parts  1 


ALTERNATE. 


277 


CASE   I. 

268.   To  find  the  2'n'02'>ortional  2^0 fts. 
1.  A  miller  Avould  mix  wheat,  worth   12  shillings  a  bushel; 
corn,  worth  8   shillings  ;  and  oats,  worth  5  shillings,  so  as  to 
make  a  mixture  worth  7  shillings  a  bushel :  what  are  the  propor- 
tional parts  of  each  ? 


OPERATION. 


oats, 


75. 


5  s.-. 
8,s'.J 


corn, 
wheat,  12s, 


A. 

B. 

c. 

D. 

2^ 

i 

5 

1 
2 

J 
r, 

2 

E. 

6  or 

2    " 

9      a 


Analysis. — On  every  bushel  put  into  the  mixture,  whose  price  is 
less  than  the  mean  price,  there  will  be  a  gain;  on  every  bushel 
whose  price  is  greater  than  the  mean  price,  there  will  be  a  loss  ;  and 
since  there  is  to  be  neither  gain  nor  loss  by  the  mixture,  the  gains 
and  losses  must  balance  each  other. 

A  bushel  of  oats,  when  put  into  the  mixture,  will  bring  7  shillings, 
giving  a  gain  of  2  shillings;  and  to  gain  1  shilling,  we  mast  take 
half  as  much,  or  |-  a  bushel,  which  we  write  in  column  A. 

On  1  bushel  of  wheat  there  will  be  a  loss  of  5  shillings  ;  and  to 
make  a  loss  of  1  shilling,  W"c  must  take  -^  of  a  bushel,  which  we 
write  in  column  A  :  \  and  -J-  are  called  proportional  numbers. 

Again  :  comparing  the  oats  and  corn,  there  is  a  gain  of  2  shillings 
on  every  bushel  of  oats,  and  a  loss  of  1  shilling  on  every  bushel  of 
corn :  to  gain  1  shilling  on  the  oats,  and  lose  1  shilling  on  the  corn, 
we  must  take  \  a  bushel  of  the  oats,  and  1  bushel  of  the  corn  :  these 
numbers  are  written  in  column  B.  Two  simples,  thus  compared,  are 
called  a  couplet :  in  one.  the  price  of  unity  is  less  than  the  mean  price, 
and  in  the  other  it  is  greater. 

If,  every  time  we  take  i  a  bushel  of  oats  we  take  -g-  of  a  bushel 
of  W'heat,  the  gain  and  loss  will  balance ;  and  if  every  time  we  take 
i  a  bushel  of  oats  we  take  1  bushel  of  corn,  the  gain  and  loss  will 
balance  :  hence,  if  the  proportional  numbers  of  a  couplet  be  multiplied 
bv  any  number^  the  gain  and  loss  denoted  by  the  products  xcill  balance. 

When  the  proportional  numbers,  in  any  column,  are  fractional 
(as  in  columns  A  and  B),  multiply  them  by  the  least  common  mul- 
tiple of  their  denominators,  and  write  the  products  in  new  cohimns 
C  and  D.  Then,  add  the  numbers  in  columns  C  and  D,  standing 
opposite  each  simple,  and  if  their  sums  have  a  common  factor,  reject 
it:  the  last  result  will  be  the  jiro;  urtional  luunbers. 

l:{ 


278  ALLIGATION. 

Note. — The  answers  to  the  last,  and  to  all  similar  questions,  will 
be  infinite  in  number,  for  t-\TO  reasons  : 

1st.  If  the  proportional  numbers  in  column  E  be  multiplied  by 
any  number,  integral  or  fraetional,  the  products  will  denote  propor- 
tional parts  of  the  simples. 

2d.  If  the  proportional  numbers  of  aiiy  covplct  be  multiplied  by 
any  number,  the  gain  and  loss  in  that  couplet  will  still  balance,  and 
the  proportional  numbers  in  the  final  result  will  be  clianged. 

Rule. — I.  Write  the  prices  or  qualities  of  the  simples  in  a 
column,  beginning  ivith  the  loivest,  and  the  mean  p>r ice  or  qualitij 
at  the  left. 

II.  Opposite  the  first  sim2)le  write  the  part  ichich  must  be 
taken  to  gain  1  <f  the  mean  j^^'icc^  und  opposite  the  other  simple 
of  the  couplet  write  the  2)ni't  which  must  be  taken  to  lose  1  of  the 
mean  price,  and  do  the  same  for  each  simple. 

III.  When  the  prop)ortional  numbers  are  fractional,  reduce 
them  to  integral  numbers,  and  then  add  those  lohich  stand  oppo- 
site the  same  simj^le :  if  the  sums  have  a  common  factor,  reject 
it :   the  result  will  denote  the  proportional  parts. 

EXA5IPLES. 

1.  TVIiat  proportions  of  coffee,  at  8  cents,  10  cents,  and  14 
cent.s  per  pound,  must  be  mixed  together  so  that  the  compound 
shall  be  worth  12  cents  per  pound  ? 

2.  A  merchant  has  teas  worth  40  cents,  65  cents,  and  75 
cents  a  pound,  from  Avhich  lie  wishes  to  make  a  mixture  worlh 
60  cents  a  pound  :  what  is  the  smallest  quantity  of  each  tliat 
he  can  take  and  express  the  parts  by  whole  numbers  ? 

3.  A  farmer  sold  a  number  of  colts  at  $50  each,  oxen  at  $10, 
cows  at  $25,  calves  at  $10,  and  realized  an  average  price  of 
$30  per  head  :  what  was  the  smallest  number  he  could  sell  of 
each  ? 

4.  What  is  the  smallest  quantity  of  water  that  must  bo 
mixed  wiili  wine  wordi  14.v.  and  15s.  a  gallon,  to  Ibrni  a  mix- 
ture worlh  i;3s.  a  gallon,  when  all  the  parts  arc  expi'csscd  by 
whole  nuuibci'a? 


ALTEKNATK, 


279 


CASE    II. 

269.    Whe7i  the  qu  inliti/  of  one  of  the  simples  is  given. 

1.  A  farmer  would  mix  rye  woi-th  80  cents  a  bushel,  and 
corn  worth  75  cents  a  bushel,  witli  GG  bushels  of  oats  worth  45 
cents  a  bushel,  so  that  the  mixture  shall  be  worth  50  cents  a 
bushel :  how  much  must  be  taken  of  each  sort  ? 

OPERATION. 


A. 

B. 

c. 

D. 

E. 

F. 

1 

] 

5 

G 

5 

11 

GG 

1 

1 

1 

6 

1 
ao 

25 

1 

1 

6 

Analysis. — Find  the  proportional  parts,  as  in  Case  I:  they  are  11, 
1  and  1.  But  we  are  to  take  66  bus-bcls  of  oats  in  the  mixture: 
hence,  each  proportional  number  is  to  be  taken  6  times ;  that  is,  as 
many  times  as  there  are  units  in  the  quotient  of  66  -r  11. 

Rule. — I.  Find  the  'proportional  numbers  as  in  Case  I,  and 
write  the  given  simjiles  opposite  its  2>i'oportional  number. 

II.  Multiply  the  given  simple  by  the  ratio  tvhich  its  propor- 
tional number  bears  to  each  of  the  others,  and  the  products  will 
denote  the  quantitits  to  be  taken  of  each. 

Note. — If  we  multiply  the  numbers  in  either  or  both  of  the 
columns  C  or  D  by  any  number,  the  proportion  of  the  numbers  in 
column  E  will  be  changed.  Thus,  if  we  multiply  column  D  by  12, 
we  shall  have  60  and  12,  and  the  numbers  in  column  E  become 
66,  12  and  1,  numbers  which  will  fulfil  the  conditions  of  the  question. 

EXAMPLES. 

1.  What  quantity  of  teas  at  12*-.  \Qs.  and  Gv.  must  be  mixed 
with  20  pounds,  at  4s.  a  pound,  to  make  the  mixture  worth  Ss. 
a  pound  ? 

2.  How  many  pounds  of  sugar,  at  7  cents  and  11  cents  a 
pound,  must  be  mixed  with  75  pounds,  at  12  cents  a  pound,  so 
that  the  mixture  may  be  worth  10  cents  a  pound  ? 


269.  How  do  you  find  the  proportional  parts  when  the  quantity  of  one 
simple  is  ijivcii  ! 


280 


ALLIGATION. 


3.  How  many  gcallons  of  oil,  at  7s.,  7s.  Qd.  and  9s.  a  gallon, 
mu>t  be  mixed  v.itli  24  gallons  of  oil,  at  9^-.  Gd.  a  gallon,  so  as 
to  form  a  mixture  worth  8s.  a  gallon  ? 

4.  Bought  10  knives  at  $2  each  :  how  many  must  be  bought 
at  §1  each,  that  the  average  price  of  the  whole  shall  be 
m? 

0.  A  grocer  mixed  50/^.  of  sugar  worth  10  cents  a  pound, 
with  sugars  worth  9i  cents,  7-i  cents,  7  cents,  and  5  cents  a 
pound,  and  found  the  mixture  to  be  worth  8  cents  a  pound : 
how  much  did  he  take  of  each  kind  ? 

CASE   III. 

270.    When  the  quantity  of  the  mixture  is  given. 

1.  A  silversmith  has  four  sorts  of  gold,  viz.,  of  24  carats  fine, 
of  22  carats  fine,  of  20  carats  fine,  and  of  15  carats  fine :  he 
would  make  a  mixture  of  42  ounces  of  17  carats  fine  :  how 
much  must  he  take  of  each  sort  ? 

OPERATION. 


17  ^ 


A. 

B. 

C. 

D. 

E. 

F. 

G. 

H. 

"I'^nO 

1 

2 

I 

2 

7 

5 

o 

15 

30 

20J 

1 

2 

2 

4 

22  ) 

1 

5 

3 

2 

2 

4 

24— 

_y 

] 

7 

2 

2 

4 

Proportional  Parts  : 
15  +  2  +  2  +  2  =  21;     42 -+21  =  2. 

Rule. — I.  Find  the  proportional  parts  as  in  Case  I. 

II.  Divide  the  quantity  of  the  mixture  by  the  sum  of  the  pro- 
portional jmrts,  and  the  quotient  will  denote  hoio  many  times 
each  part  is  to  be  talceii.  Multiply  this  quotient  by  the  parts 
separately,  and  each  2>yoduct  tvill  denote  the  quantity  of  the  cor- 
resjiondiiig  simple. 


270.   How  do  you   liiul  ilir  proportional  i)arls  wlieii  llic  iiiiaiitity  of  llif 
mixUirc  is  j^'ivcii  1 


ALLIGATION.  281 

Note. — We  may,  as  in  the  other  cases,  multiply  each  couplet  by 
any  number  we  please,  which  will  merely  change  the  relation  of  the 
proportional  parts,  and  consequently  uivc  diirerent  proportions  of  the 
ingredients.  Hence,  there  is  an  infinite  number  of  answers^  if  we 
employ  fractions,  and  often  many  answers  to  similar  questions,  in 
whole  numbers.* 

EXAMPLES. 

1.  A  grocer  has  teas  at  5s.,  Os.,  8?.,  and  Os.  a  pound,  and 
wislies  to  make  a  compound  of  88/i.  worth  7^.  a  pound  :  how 
much  of  each  sort  must  be  taken  ? 

2.  A  liquor  dealer  wishes  to  fill  a  hogshead  with  water,  and 
two  kinds  of  brandy,  at  $2,50  and  $3,00  per  gallon,  so  that  the 
mixture  may  be  worth  $2,25  a  gallon  :  in  what  proportions 
must  he  mix  them  ? 

3.  A  person  sold  a  number  of  sheep,  calves,  and  lambs,  40 
in  all,  for  $48  :  how  many  did  he  sell  of  each,  if  he  received 
for  each  calf  $lf,  each  sheep  111,  and  each  lamb  If? 

4.  A  merchant  sold  20  stoves  for  8180  ;  for  the  largest  size 
he  received  $19  each,  for  the  middle  size,  $7,  and  for  the  small 
size  %Q:  how  many  did  he  sell  of  each  kind  ? 

5.  A  vintner  has  wines  at  45.,  65.,  8s.,  and  IO5.  per  gallon  > 
he  wishes  to  make  a  mixture  of  120  gallons,  Avorth  os.  per  gal- 
lon :  what  quantity  must  he  take  of  each  ? 

6.  A  tailor  has  24  garments,  worth  |;144.  He  has  coats, 
pantaloons  and  vests,  worth  $12,  $5  and  $2  each,  respectively : 
how  many  has  he  of  each  ? 

7.  A  jeweler  melted  together  four  sorts  of  gold,  of  24,  22, 
20  and  15  carats  fine,  so  as  to  produce  a  compound  of  42o2'.  of 
17  carats  fine  :  how  much  did  he  take  of  each  sort  ? 

8.  A  man  paid  $70  to  3  men  for  35  days  labor :  to  the  first 
lie  paid  |5  a  day,  to  the  second,  $1  a  day,  and  to  the  third,  $^ 
a  day  :  how  many  days  did  each  labor  ? 

*  See   an  admirable   article   on  Alhgation,   published  by  Professor  D, 

Wood,  in  the  June  number  of  the  New  ifork  Teacher  for   1855.     B\'  his 

permission,  I  have  used  such  parts  of  it  as   seemed  appropriate  to  a  Text 

Book. 

1:! 


282  COIN'S    AND    CUEEEXCIES. 

COINS   AND    CURRENCIES. 

271.  Coins  ai-e  pieces  of  metal,  of  gold,  silver,  or  copper, 
of  fixed  values,  and  impressed  with  a  public  stamp  prescribed 
by  the  country  where  they  are  made.  These  are  called  specie, 
and  are  generally  declared  to  be  a  legal  tender  in  payment  of 
debts.  The  Constitution  of  the  United  States  provides,  that 
gold  and  silver  only,  shall  be  a  legal  tender. 

The  coins  of  a  country,  and  those  of  foreign  countries  having 
a  fixed  value  established  by  law,  together  with  bank  notes 
redeemable  in  specie,  make  up  what  is  called  the  Currency. 

272.  A  Foreign  coin  may  be  said  to  have  four  values  : 

1st.  The  intrinsic  value,  which  is  determined  by  the  amount 
of  pure  metal  which  it  contains  : 

2d.  The  Custom  House  or  legal  value,  which  is  fixed  by  law  : 

3d.  The  mercantile  value,  which  is  the  amount  it  will  sell  for 
in  open  market : 

4th.  The  Exchange  value,  which  is  the  value  assigned  to  it 
in  buying  and  selling  bills  of  exchange  between  one  country 
and  another. 

Let  us  take,  as  an  example,  the  English  pound  sterling, 
which  is  represented  by  the  gold  sovereign.  Its  intrinsic  value, 
as  determined  at  the  Mint  in  Philadelphia,  compared  with  our 
gold  eagle,  is  $4,861.  Its  legal  or  custom  house  value  is  $4,84. 
Its  commercial  value,  that  is,  what  it  will  bring  in  Wall-street, 
New  York,  varies  from  $4,83  to  $4,80,  seldom  reaching  either 
the  lowest  or  highest  limit.  The  excliange  value  of  the  Eng- 
lish pound,  is  $4,44|-,  and  was  the  legal  value  before  the  change 
in  our  standard.  This  change  raised  the  legal  value  of  the 
pound  to  $4,84,  but  merchants  and  dealers  in  exchange  pre- 
ferred to  retain  the  old  value,  which  became  nominal,  and  to 
add  the  difference  in  the  form  of  a  iiremium  on  excInnujCy 
which  is  cx]ilained  in  Art.  •287.  For  the  values  of  the  various 
coins,  see  Table,  page  3'Jl. 


271.   Wliat  are  <!i»ins  !     Wliat   arc   tlipv  callpin     What  is  dnclarocl  in 


EXCHANGE.  283 

EXCHANGE. 

273.  Exchange  is  a  term  which  denotes  the  payment  of 
money  by  a  person  residing  in  one  place  to  a  person  residing  in 
another.  The  payment  is  generally  made  by  means  of  a  bill 
of  exchange. 

274.  A  Bill  of  Exchange  is  an  open  letter  of  request 
from  one  person  to  another,  desiring  the  payment  to  a  third 
party  named  therein,  of  a  certain  sum  of  money  to  be  paid  at  a 
specified  time  and  place.  There  are  always  three  parties  to  a 
bill  of  exchange,  and  generally  four  : 

1.  He  who  writes  the  open  letter  of  request,  is  called  the 
draiver  or  maker  of  the  bill : 

2.  The  person  to  whom  it  is  directed  is  called  the  drawee  : 

3.  The  person  to  whom  the  money  is  ordered  to  be  paid  is 
called  the  payee  ;  and 

4.  Any  person  who  purchases  a  bill  of  exchange  is  called  the 
buyer  or  remitter. 

275.  Bills  of  exchange  are  the  proper  money  of  commerce. 
Suppose  Mr.  Isaac  Wilson  of  the  city  of  N'ew  York,  ships  1000 
bags  of  cotton,  worth  £G000,  to  Samuel  Johns  &  Co.  of  Liver- 
pool ;  and  at  about  the  same  time  William  James  of  New  York 
orders  goods  from  Liverpool,  of  Ambrose  Spooner,  to  the  amount 
of  six  thousand  pounds  sterling.  Now,  Mr.  Wilson  draws  a 
bill  of  exchange  on  Messrs.  Johns  &  Co.  in  the  following 
form,  viz. : 

regard   to   them  ?     "N^'hat   is  provided  by  the  Constitution  of  the  United 
States  1     What  do  you  understand  by  Currency  1 

272.  How  many  values  may  a  coin  be  said  to  have  1  What  is  the  in- 
trinsic vakie  1  ^Vhat  is  its  Custom  House  value  \  What  is  the  niercantilu 
vahie  !     What  is  tl^e  exchange  value  ? 

273.  What  is  Exchange  ^     How  is  the  payment  generally  made  ? 

274.  What  is  a  Bill  of  Exchange  1  How  many  parties  are  there  to  u 
Dill  of  exchange  1     Name  them. 

275.  How  do  bills  of  exchange  aid  commerce"!  Vame  all  the  parties  to 
the  bill  in  this  example. 


284  EXCHANGE. 


Exchange  for  £G000.  New  York,  July  30th,  1846- 

Sixty  days  after  sight  of  this  my  first  Bill  of  Exchange 
(second  and  third  of  the  same  date  and  tenor  unpaid*)  pay  to 
David  C.  Jones  or  order,  six  thousand  pounds  sterling,  with 
or  without  further  notice.  Isaac  Wilson. 

3Iessrs.  Samuel  Johns  &  Co., ) 
Merchants,  Liverpool.        ) 

Let  us  now  suppose  that  Mr.  James  j^urchases  this  bill  of 
David  C.  Jones  for  the  purpose  of  sending  it  to  Ambrose 
Spooner  of  Liverpool,  whom  he  owes.  We  shall  then  have 
all  the  parties  to  a  bill  of  exchange ;  viz.,  Isaac  Wilson,  the 
maker  or  drutcer ;  Messrs.  Johns  &  Co.,  the  drawees ;  David 
C.  Jones,  the  payee ;  and  William  James,  the  buyer  or  re- 
mitter. 

276.  A  bill  of  exchange  is  called  an  inland  hill,  when  the 
drawer  and  drawee  both  reside  in  the  same  countiy ;  and  when 
they  reside  in  different  countries,  it  is  called  a  foreign  hill. 
Thus,  all  bills  in  which  the  drawer  and  drawee  reside  in  the 
United  States,  are  inland  bills  ;  but  if  one  of  them  resides  in 
England  or  France,  the  bill  is  a  foreign  bill. 

277.  The  time  at  which  a  bill  is  made  payable  varies,  and  is 
a  matter  of  agreement  between  the  drawer  and  buyer.  Tliey 
may  either  be  drawn  at  sight,  or  at  a  certain  number  of  days 
after  sight,  or  at  a  certain  number  of  days  after  date. 

278.  Days  of  Guace  are  a  certain  number  of  days  granted 
to  the  person  who  pays  the  bill,  after  the  time  named  in  the  bill 

276.  ^^'hat  is  an  inland  bill  !  What  is  a  foreign  bill  ?  Arc  bills  drawn 
between  one  state  and  another  inland,  or  foreign  1 

277.  How  is  the  time  determined  at  which  a  bill  is  made  payable  1  How 
are  bills  always  drawn  1 

*  Three  bills  are  generally  drawn  for  the  same  amount,  called  the  first, 
second,  and  third,  and  together  they  form  a  set.  One  only  is  paid,  and 
then  the  oUier  two  are  of  no  value.  This  arranjrement  avoids  the  acci- 
dents  and  delays  incident  to  transmiuinr;  the  bills. 


KXCIIANGE.  285 

has  expired.     In  the  United   States  and  Great   Bi-itain  three 
days  are  allowed. 

279.  In  ascertaining  the  time  when  a  bill  payable  so  many 
days  after  sight,  or  ai'ter  date,  actually  falls  due,  the  day  of 
presentment,  or  the  day  of  the  date,  is  not  reckoned.  When 
the  time  is  expressed  in  months,  otlendar  monlha  are  always 
understood. 

If  the  month  in  which  a  bill  falls  due  is  shorter  than  the  one 
hi  which  it  is  dated,  it  is  a  rule  not  to  go  on  into  the  next 
month.  Thus,  a  bill  drawn  on  the  28th,  29th,  30th,  or  31st  of 
December,  payable  two  months  after  date,  would  fall  due  on 
the  last  of  February,  except  for  the  days  of  grace,  and  would 
be  actually  due  on  the  third  of  March. 

ENDORSING    BILLS. 

280.  In  exaniining  the  bill  of  exchange  drawn  by  Isaac 
Wilson,  it  will  be  seen  that  Messrs.  Johns  &  Co.  are  requested 
to  pay  the  amount  to  David  C.  Jones  or  order  ;  that  is,  either 
to  Jones  or  to  any  other  person  named  by  him.  If  Mr.  Jones 
simply  writes  his  name  on  the  back  of  the  bill,  he  is  said  to 
endorse  it  in  blank;  and  the  drawees  must  pay  it  to  any  rightful 
owner  who  presents  it.  Such  rightful  owner  is  called  the  holder, 
and  Mr.  Jones  is  called  the  endorser. 

If  Mr.  Jones  writes  on  the  back  of  the  bill,  over  his  signa- 
ture, "  Pay  to  the  order  of  William  James,"  this  is  called  a 
special  endorsement,  and  William  James  is  the  endorsee,  and  he 
may  either  endorse  in  blank  or  write  over  his  signature,  "  Pay 

27S.  ^Vllat  are  days  of  grace  1  How  many  days  of  grace  are  allowed 
hi  this  country  and  in  Great  Britain  ! 

279.- In  ascertaining  the  time  when  a  bill  is  payable,  what  days  are 
reckoned  \  When  the  time  is  expressed  in  months,  wbat  kind  of  months 
is  under.stood  !  If  the  month  in  which  tlie  bill  falls  due  is  shorter  than 
that  ill  which  it  is  drawn,  what  rule  i.?  observed  1 

280.  What  is  an  cndor.scmcnt  in  blank  1  What  is  the  person  making  il 
called  '  What  is  a  special  endorsement  !  What  is  tl;c  effect  of  an  en- 
dorsement '      How  may  a  bill  drawn  to  bearer  be  transft.'rred  .' 


28()  KXCHANGK, 

to  tlie  order  of  Ambrose  Spooner,"  and  the  drawees,  Messrs. 
Johns  &  Co.,  will  then  be  bound  to  pay  ihe  amount  to  Mr. 
Spooner. 

A  bill  drawn  payable  to  bearer,  may  be  transferred  by  Ti:';re 
delivery. 

ACCEPTANCE. 

281.  When  the  bill  drawn  on  Messrs.  Jolms  &  Co.  is  pre- 
sented to  them,  they  must  inform  the  holder  whether  or  not  they 
will  pay  it  at  the  expiration  of  the  time  named.  Their  agree- 
ment to  pay  it  is  signified  by  writing  across  the  face  of  the  bill, 
and  over  their  signature  the  word  "  accepted,"  and  they  are  then 
called  the  acce2)tors. 

LIABILITIES    OF    THE    PARTIES. 

282.  The  drawee  of  a  bill  does  not  become  responsible  for 
its  payment  until  after  he  has  accepted.  On  the  presentation 
of  the  bill,  if  the  drawee  does  not  accept,  the  holder  should 
immediately  take  means  to  have  the  drawer  and  all  the  en- 
dorsers notified.  Such  notice  is  called  a  pro/esi,  and  is  given 
by  a  public  officer  called  a  notary,  or  notary  public.  If  the 
parties  are  not  notified  in  a  reasonable  time,  they  are  not  respon- 
sible for  the  payment  of  the  bill. 

If  the  drawer  accepts  the  bill,  and  fails  to  make  the  payment 
when  it  becomes  due,  the  parties  must  be  notified  as  before,  and 
this  is  called  protcsti/iy  the  bill  for  non-payment.  If  the  en- 
dorsers are  not  notified  in  a  reasonable  time,  they  are  not 
responsible  for  the  amount  of  the  bill. 


281.  ^A'llat  is  an  arcrptancc  !     How  is  it  inado  ] 

282.  ^\'ll(n)  (Iocs  the  drawee  of  a  bill  become  rc.spon.sible  for  its  pay- 
ment! If  the  drawee  docs  not  accept,  what  nni.st  tlie  holder  do!  What 
is  .such  notice  called  !  13y  whom  is  it  made!  If  the  parties  to  the  bill 
are  not  notified,  what  is  the  consequence  !  If  the  drawee  accepts  the  bill 
and  fails  to  make  Ihe  payment,  what  must  then  be  done  !  If  the  bill  if 
not  protested,  what  will  be  the  consequence  ! 


EXCHANGE.  287 

PAR  OF  EXCHANGE — COURSE  OF  EXCHANGE. 

283.  The  intrinsic  ixtr  of  exchange,  is  a  term  used  to  com- 
pare tlie  coins  of  different  countries  with  each  other,  witli  respect 
to  their  intrinsic  values,  that  is,  with  reference  to  the  amount  of 
pure  metal  in  eacli.  Tims,  the  English  sovereign,  which  repre- 
sents the  pound  sterling,  is  intrinsically  worth  $4,8G1  in  our 
gold,  taken  as  a  standard,  as  determined  at  the  Mint  in  Phila- 
delphia. This,  therefore,  is  the  value  at  which  the  sovereign 
must  be  reckoned,  in  estimating  the  par  of  exchange. 

284.  The  commercial  jxn-  of  exchanr/e  is  a  comparison  of  the 
coins  of  different  countries  according  to  their  market  value 
Tiius,  as  the  market  value  of  the  English  sovereign  varies  from 
$4,8.j  to  $4,85  (Art.  272),  the  commercial  par  of  exchange 
will  fluctuate.  It  is,  however,  always  determined  when  we 
know  the  value  at  which  the  foreign  coin  sells  in  our  market. 

285.  The  course  of  exchange  is  the  variable  price  which  is 
paid  at  one  place  for  bills  of  exchange  drawn  on  another.  The 
course  of  exchange  difiers  from  the  intrinsic  par  of  exchange, 
and  also  from  the  commercial  par,  in  the  same  way  that  the 
market  price  of  an  article  differs  from  its  natural  price.  Tlie 
commercial  par  of  exchange  would  at  all  times  determine  the 
course  of  exchange,  if  there  were  no  fluctuations  in  trade. 

286.  When  the  market  price  of  a  foreign  bill  is  above  the 
commercial  par,  the  exchange  is  said  to  be  at  a  premium,  or  in 
favor  of  the  foreign  place,  because  it  indicates  that  the  foreign 

283.  What  do  you  understand  by  the  intrinsic  par  of  exchange  1  \A'hat 
is  the  intrinsic  value  of  the  English  sovereign  1 

284.  What  is  the  commercial  par  of  exchange  \  What  is  the  comnicr 
cial  value  of  the  English  sovereign  1 

285.  What  do  you  understand  by  the  course  of  exchange  ?  How  does 
it  differ  from  the  intrinsic  par  and  the  commercial  par  !  What  causes  it  to 
differ  from  the  commercial  par  ! 

286.  ^^'hat  is  said  when  the  price  of  a  foreign  bill  is  above  the  commer- 
cial par  1  When  is  it  below  it  ?  To  whom  is  a  favorable  state  of  exchange 
advantageous  1     To  whom  is  it  injurious  \ 


288  EXCHANGE. 

place  has  sold  more  than  it  has  bought,  and  that  specie  must  be 
shipped  to  make  up  the  difference.  "When  the  market  price  is 
beloio  this  par,  exchange  is  said  to  be  below  par,  or  in  favor  of 
the  place  where  the  bill  is  drawn.  Such  place  will  then  be  a 
creditor,  and  the  debt  must  be  paid  in  specie  or  other  property. 
It  should  be  observed  that  a  favorable  state  of  exchange  is 
advantageous  to  the  buyer  but  not  to  the  seller,  whose  interest, 
as  a  dealer  in  exchange,  is  identified  with  that  of  the  place  on 
which  the  bill  is  drawn. 

287.  It  was  stated  in  Art.  272,  that  the  exchange  value  of 
the  pound  sterling  is  $4,44^  —  4,4444  4-  ;  that  is,  this  value  is 
the  basis  on  \\  liich  the  bills  of  exchange  are  drawn.  Now  this 
value  being  below  both  the  commercial  and  intrinsic  value,  the 
drawers  of  bills  increase  the  course  of  exchange  so  as  to  make 
up  this  deficiency. 

For  example,  if  we  add  to  the  exchange  value  of  the  pound, 
9  per  cent,  we  shall  have  its  commercial  value,  very  nearly. 
Thus,  exchange  value,  -         -         ==  $4,4444  + 

Nine  per  cent,     ----:=      ,3999  + 
which  gives       -         -         -  $4,8443 

and  this  is  tlie  average  of  the  commercial  value,  very  nearly. 
Therefore,  when  the  course  of  exchange  is  at  a  premium  of 
9  per  cent,  it  is  at  the  commercial  par,  and  as  between  England 
and  this  country  it  would  stand  near  this  point,  but  for  the  fluc- 
tuations of  trade  and  other  accidental  circumstances. 

INLAND  BILLS. 

288.  We  have  seen  that  inland  bills  are  those  in  which  the 
drawer  and  drawee  both  reside  in  the  same  country  (Art.  276), 

EXAMI'LES. 

1.  A  merchant  at  New  Orleans  wishes  to  remit  to  New  York 
$8405,  and  exchange  is  1^  per  cent  premium.  How  much 
must  he  pay  for  such  a  bill  ? 


287.  What  is  the  exchange  value  of  the  pound  slerhng  1 

288.  What  are  inland  bills  1 


EXCHANGE.  289 

2.  A  merchant  in  Boston  wishes  to  pay  in  Phihulelphia 
$8746,50  ;  exchange  between  Boston  and  Philadelphia  is  1}  per 
cent  below  par.     What  must  he  pay  for  a  bill  ? 

0.  A  merchant  in  Philadelphia  wishes  to  pay  $9876,40  in 
Baltimore,  and  finds  exchange  to  be  1  per  cent  below  par  :  what 
must  he  pay  for  the  bill  ? 

ENGLAND. 

289.  It  has  already  been  stated  that  llie  exchanges  between 
this  country  and  England  are  made  in  pounds,  shillings,  and 
pence,  and  that  the  exchange  value  of  the  pound  sterling  is 
$4,44|-,  and  tliat  the  premiums  are  all  reckoned  from  this 
standard, 

EXAMPLES. 

1.  A  merchant  in  New  York  wishes  to  remit  to  Liverpool 
£11 G7  10s.  6c?.,  exchange  being  at  8i  per  cent  premium. 
How  much  must  he  pay  for  the  bill  in  Federal  money  ? 

First,  £1167  10s.  6c?.  -         -         =£1167.525 

For  multiply  by  8^  per  cent,         -  .085 

the  product  is  the  premium  -         -  =  99.239625 

the  product  added  gives         -         -  £1266.764625 

which  reduced  to  dollars  and  cents  at  the  rate  of  $4,44f  to  the 
pound,  gives  the  amount  which  must  be  paid  for  the  bill  in 
dollars  and  cents. 

2.  A  m.erchant  has  to  remit  £36794  8s.  dd.  to  London,  how 
much  must  he  pay  for  a  bill  in  dollars  and  cents,  exchange  being 
7|-  per  cent  premium  ? 

3.  A  merchant  in  New  York  wishes  to  remit  to  London 
$67894,25,  exchange  being  at  a  premium  of  9  per  cent.  What 
will  be  the  amount  of  his  bill  in  pounds  shillings  and  pence  ? 

Note. — Add  the  amount  of  the  premium  to  the  exchange  value  of 
the  pound,  viz.  S4,44f ,  which  in  this  case  gives  $4.84444,  and  then 
divide  the  amount  in  dollars  by  ihis  sum,  and  the  quotient  will  bo 
tlie  amount  of  the  bill  in  pounds  and  the  decimals  of  a  pound. 


289.  In   what   currency  are   the   excManges  lietween  this  country   and 
England  made  ?     A\'hat  is  the  exchange  value  of  the  pound  sterling  1 


290  EXOHANUE. 

4.  A  mercbant  in  New  York  owes  £1256  18s.  9c?.  in  London  ; 
exchange  at  a  nominal  premium  of  l-h  per  cent :  how  much 
money,  in  Federal  currency,  will  be  necessary  to  purchase  the 

bill? 

5.  I  have  S947,8G  and  wish  to  remit  to  London  £364  IBs.  8a'., 
exchange  being  at  8i  per  cent :  how  much  additional  money 
will  be  necessary  ? 

FRANCE. 

290.  The  accounts  in  France,  and  the  exchange  between 
France  and  other  countries,  are  all  kept  in  francs  and  centimes, 
■which  are  hundredths  of  the  franc.  We  see  from  the  table  that  tht 
value  of  the  franc  is  18.6  cents,  which  gives  very  nearly,  5  francs 
and  38  centimes  to  the  dollar.  The  rate  of  exchange  is  com- 
puted on  the  value  18.6  cents,  but  is  often  quoted  by  stating  the 
value  of  the  dollar  in  francs.  Thus,  exchange  on  Paris  is  said 
to  be  5  francs  40  centimes,  that  is,  one  dollar  will  buy  a  bill  on 
Paris  of  5  francs  and  40  hundreds  of  a  franc. 

EXAMPLES. 

1.  A  merchant  in  New  York  wishes  to  remit  167556  francs 
to  Paris,  exchange  being  at  a  premium  of  11  per  cent.  What 
will  be  the  cost  of  his  bill  in  dollars  and  cents  ? 

Commercial  value  of  the  franc,  -         -         18.6  cents, 
Add  li  per  cent,      -         -         -         -  .279 

Gives  value  for  remitting,  -         -         18.879  cents; 

then,     167556  x  18.879  :=  $31632,89724, 
which  is  the  amount  to  be  paid  for  the  bill  ? 

2.  What  amount,  in  dollars  and  cents,  will  purchase  a  bill  on 
Paris  for  86978  francs,  exchange  being  at  the  rate  of  5  francs 
and  2  centimes  to  the  dollar  ? 

First,     86978  -^  5.02  =  $17326,274  +,  the  amount. 
Is  this  bill  above  or  below  par  ?     What  per  cent  ? 

290.  In  what  currency  are  the  exchanges  with  France  conducted^ 
What  is  a  centime  ?  "What  is  the  vahie  of  a  franc  I  \\hi\l  is  meant  Then 
exchange  on  Paris  is  quoted  at  5  francs  40  centimes  1 


EXCHANGE.  291 

3.  How  much  money  must  be  paid  to  purchase  a  bill  of  ex- 
change on  Paris  for  G8097  francs,  exchange  being  3  per  cent 
below  par  ? 

4.  A  merchant  in  New  York  wishes  to  remit  $16785,25  to 
Paris ;  exchange  gives  5  francs  4  centimes  to  the  dollar :  how 
much  can  he  remit  in  the  currency  of  Paris  ? 

HAMBURG. 

291.  Accounts  and  exchanges  with  Hamburg  are  generally 
made  in  the  marc  banco,  valued,  as  we  see  in  the  tabic,  at 
35  cents. 

EXAMPLES. 

1.  "What  amount  in  dollars  and  cents  will  purchase  a  bill  of 
exchange  on  Hamburg  for  18649  marcs  banco,  exchange  being 
at  2  per  cent  premium  ? 

2.  What  amount  will  purchase  a  bill  for  3678  marcs  banco, 
reckoning  the  exchange  value  of  the  marc  banco  at  34  cents  ? 
Will  this  be  above  or  below  the  par  of  exchange  ? 


ARBITRATION   OF   EXCHANGE. 

292.  Arbitration  of  Exchange  is  the  method  by  which  the 
currency  of  one  country  is  changed  into  that  of  another,  through 
the  medium  of  one  or  more  intervening  currencies,  with  which 
the  first  and  last  are  compared. 

293.  When  there  is  but  one  intervening  currency  it  is  called 
Simple  Arbitration  ;  and  when  there  is  more  than  one  it  is  called 
Conqiound  Arbitration.  The  method  of  performing  this  is  called 
the  Chain  Rule. 

291  In  what  are  accounts  kept  at  Hamburgh  1  What  is  the  value  o/ 
the  marc  banco  1 

292.  What  is  arbitration  of  exchange  ?    . 

293.  When  there  is  but  one  intervening  currency,  what  is  the  exchange 
called  1     When  there  is  more  than  one,  what  is  it  called  1 


232  AEBITKATION    OK   EXCHANGE. 

29  i.  The  pi'inciple  involved  iii  arbitration  of  exchange  is 
simply  this  :  To  pass  from  one  system  of  values  through 
several  others,  and  find  the  true  proportion  between  the  first 
and  last. 

1.  Suppose  it  were  required  to  exchange  109150  pence  into 
dollars,  by  first  changing  them  into  shillings,  then  into  pounds, 
and  then  into  dollars. 

OPERATION. 

109150^.  =  109150  X  j\s. 
109150^.  =  £109150  X  Jg-  X  2V 
109150(;.  =  $109150  X  Jj  X  Jo  X  4-444  =  $201,924. 

Analysis. — Since  12  pence  make  1  shilling,  there  will  be  one- 
twelfth  as  many  shillings  as  pence  :  since  20  shillings  make  1  pound, 
there  will  be  one-twentieth  as  many  pounds  as  shillings ;  and  since 
there  are  ^4.444  in  a  pound,  there  will  be  as  many  dollars  as  result 
from  taking  the  pounds,  4.444  times;  that  is,  S201.924. 

2.  Let  it  be  required  to  remit  $G570  to  London,  by  the  way 
of  Paris,  excliange  on  Paris  being  5  francs  15  centimes  for 
$1,  and  the  exchange  from  Paris  to  London,  25  francs  and  80 
centimes  for  £1  :  what  will  be  the  value  of  the  remittance  at 
London  ? 

OPERATION. 

$6570  =  G570  x  5.15  francs. 
$6570  =  $6570  x  5.15  x  j^'-r-q  =  =£131  3^.  lOfc/. 

Analysis. — Since  5  francs  and  15  centimes  are  equal  to  Si.  there 
■will  be  as  many  francs  at  Paris  as  are  equal  to  6570  taken  5.15 
times  ;  and  since  £l  at  London  is  equal  to  25.80  francs,  there  will 
be  as  many  pounds  as  25.80  is  contained  times  in  the  last  product; 
that  is,  jCl31  lis.  10|ri.  Hence,  the  following,  Avhich  is  called  the 
Chain  Rule  : 

Miildph/  the  Stan  to  be  remitted  by  the  foUoioina  quotienls: 
By  a  certain  amount  at  the  second  place  divided  by  its  equivalent 

2!)4.  What  principle  is  involved  in  the  arbitration  of  exchange  1  What 
is  till!  chain  rule  ?     Give  the  rule. 


AKBITRATION   OF   EXCHANGE.  293 

at  Ihe  first ;  by  a  certain  amount  at  the  third  place  by  its  equiva- 
lent at  the  second  ;  by  a  certain  amount  at  the  fourth  2'>loce  divi- 
ded by  its  equivalent  at  the  third,  and  so  on  to  the  last  place. 

EXAMPLES. 

1.  A  merchant  wishes  to  remit  $4888,40  from  New  York  to 
London,  and  the  exchange  is  at  a  premium  of  10  per  cent.  He 
finds  that  he  can  remit  to  Paris  at  5  francs  15  centimes  to  the 
dollar,  and  to  Hamburg  at  35  cents  per  marc  banco.  Now,  the 
exchange  between  Paris  and  London  is  25  francs  80  centimes 
for  £1  sterling,  and  between  Hamburg  and  London  13|-  marcs 
banco  for  £1  sterling.     How  had  he  better  remit  ? 

1st.   To  London  direct. 

The  amount  to  be  remitted  is  $4888,40,  The  exchange 
value  of  £1  is  $4,444,  and  since  the  exchange  is  at  a  premium 
of  10  per  cent,  the  value  of  £1  is  $4,444  +  ,4444  r=  $4,8884 : 
hence, 

$4888,40  X  T.sVsT  =  ^1000  : 

hence,  if  he  remits  direct  he  will  obtain  a  bill  for  £1000. 

2d.  Exchange  through  Paris. 
1.03 

4888,40  X  ^  X  j^  ^  £975,7852  =  £975  15s.  8}d. 
5.16 

Analysis. — Since  5.15  francs  are  equal  to  1  dollar,  the  first  mul- 
tiplier will  be  this  amount  divided  by  $1  ;  and  since  £l  is  equal  to 
25.80  francs,  the  second  multiplier  will  be  £^1  divided  by  this  amount. 
Then,  by  dividing  by  5  and  multiplying,  we  find  that  the  amount 
remitted  by  the  second  method  would  be  worth,  at  London,  £^975 
t5s.  S^^. 

3d.  Exchange  through  Hamburg. 

$4888,40  X  .3J3  X  13^.^5  =^1015.771  =  £1015  15s.  5d. 

Analysis. — Since  1  marc  banco  is  equal  to  35  cents,  it  is  35  hun- 
dredths of  a  dollar  :    hence,   the  first  multiplier  is   1   marc  banco 


294  ARBITRATION   OF   EXCHANGE. 

divided  by  .35,  and  the  second,  1  divided  by  13.75.     By  tlii.s  course 
of  exchange  the  remittance  at  London  would  be  worth  £1015  1 5s.  5d. 

Hence,  the  best  way  to  remit  is  through  Hamburg,  then 
direct,  and  the  least  advantageous,  through  Paris. 

2.  A  merchant  in  London  has  sold  goods  in  Amsterdam  to 
the  amount  of  824  pounds  Flemish,  which  could  be  remitted 
to  London  at  the  rate  of  34s.  4rf.  Flemish  per  pound  sterling. 
He  orders  it  to  be  remitted  circuitously  at  the  following  rates, 
viz. :  to  France  at  the  rate  of  48(/.  Flemi.sh  per  crown ;  thence 
to  Vienna  at  100  crowns  for  60  ducats  ;  thence  to  Hamburg  at 
lOOd.  Flemish  per  ducat ;  thence  to  Lisbon  at  50c/.  Flemish 
per  crusado  of  400  reas  ;  and  lastly,  from  Lisbon  to  England 
at  5.y.  8'J.  per  milrea :  does  he  gain  or  lose  by  the  circular 
exchange  ? 

48(/.  Flemish  =  1  crown, 

100  crowns  =  GO  ducats, 

1  ducat  =:  100(7.  Flemish, 

50d.  Flemish  =  400  reas, 

1  milrea  or  1000  reas   =  G8d.  sterlinjr. 


o~- 


*o- 


a:0       $        17 

1    00   100  00        ^$        824x17   14008 
^$       i00  1     00    1000  ~    25   ~   25 

$  m 

25 
=  £560  6s.  4frf. 

The  direct  exchange  would  give, 

82i  X  ^-i j:^. —  =  824  X  If"-  =  £480  sterhng. 

34s.  4c?.  Flemish  '^  **  ° 

Hence,  the  amount  gained  by  circuitous  exchange  wouliJ    b" 
£80  6s.  |rf. 


GENERAL   AVEKAGE.  295 

GENERAL  AVERAGE. 

295.  Average  is  a  term  of  commerce  and  navigation,  to 
signify  a  contribution  by  individuals,  wliere  tlie  goods  of  a  par- 
ticular merchant  are  thrown  overboard  in  a  storm,  to  save  the 
ship  from  sinking,  or  wliere  the  masts,  cables,  anchors,  or  other 
furniture  of  the  ship  are  cut  away  or  destroyed  for  the  preser- 
vation of  the  whole.  In  these  and  like  cases,  where  any  sacri- 
fices are  deliberately  made,  or  any  expenses  voluntarily  incurred, 
to  prevent  a  total  loss,  such  sacrifice  or  expense  is  the  proper 
subject  of  a  general  contribution,  and  ought  to  be  ratably 
borne  by  the  owners  of  the  ship,  the  freiglit,  and  the  cargo,  so 
that  the  loss  may  fall  proportionably  on  all.  The  amount  sacri- 
ficed is  called  the  jeidsoji. 

296.  Average  is  either  general  or  particular ;  that  is,  it  is 
either  chargeable  to  all  the  interests,  viz.,  the  ship,  the  freight, 
and  the  cargo,  or  only  to  some  of  them.  As  when  losses  occur 
from  ordinary  wear  and  tear,  or  from  the  perils  incident  to  the 
voyage,  without  being  voluntarily  incurred  ;  or  when  any  par- 
ticular sacrifice  is  made  for  the  sake  of  the  ship  only  or  the 
cargo  only,  these  losses  must  be  borne  by  the  parties  imme- 
diately interested,  and  are  consequently  defrayed  by  a  -particular 
average.  There  are  also  some  small  charges  called  pjetty  or 
accustomed  averages,  one-third  of  which  is  usually  charged  ta 
the  ship,  and  two-thirds  to  the  cargo. 

No  general  average  ever  takes  place,  except  it  can  be  shown 
that  tlte  danger  was  imminent,  and  that  the  sacrifice  was  made 
indiqoensahle,  or  sup)-posed  to  be  so  by  the  captain  and  officers, 
fur  the  safety  of  the  ship. 

297.  In  different  countries  different  modes  are  adopted  of 
valuing  the  articles  which  are  to  constitute  a  general  average. 


295.  \Miat  does  the  term  average  signify  1 

296.  How  many  kinds  of  average  are  there  !  What  are  the  small 
charges  called  1  Under  what  circumstances  will  a  general  average  take 
place 


296  GENERAT>    AVERAGE. 

In  general,  hoTvever,  the  value  of"  the  freightage  is  held  to  be 
the  clear  sura  which  the  ship  has  earned  after  seamen's  wages, 
pilotage,  and  all  such  other  charges  as  came  under  the  name 
of  petty  charges,  are  deducted ;  one-third,  and  in  some  cases 
one-half,  being  deducted  for  the  wages  of  the  crew. 

The  goods  lost,  as  well  as  those  saved,  are  valued  at  the  price 
they  Avoukl  have  brought  in  ready  money  at  tlie  j^loce  of  delivery, 
on  the  ship's  arriving  there,  freight,  duties,  and  all  other  charges 
being  deducted  :  indeed,  they  bear  their  proportions,  the  same 
as  the  goods  saved.  The  ship  is  valued  at  the  price  she  would 
bring  on  her  arrival  at  the  port  of  delivery.  But  when  the  loss 
of  masts,  cables,  and  other  furniture  of  the  ship  is  compensated 
by  general  average,  it  is  usual,  as  the  new  articles  will  be  of 
greater  value  than  the  old,  to  deduct  one-third,  leaving  two- 
thirds  only  to  be  charged  to  the  amount  to  be  contributed. 

EXAMPLES. 

1.  The  vessel  Good  Intent,  bound  from  New  York  to  New 
Orleans,  Avas  lost  on  the  Jersey  beach  the  day  after  sailing. 
She  cut  away  her  cables  and  masts,  and  cast  overboard  a  part 
of  her  cargo,  by  which  another  part  was  injured.  The  ship 
was  finally  got  off,  and  brought  back  to  New  York. 

AMOUNT    OF   LOSS. 

Goods  of  A  cast  overboard,          .         -         .  $500 

Damage  of  the  goods  of  B  by  the  jettison,   -  200 

Freight  of  the  goods  cast  overboard,    -         -  100 

Cable,  anchors,  mast,  &;c.,  worth     -     $300  )  gOQ 
Deduct  one-third,           .         -         -        100 ) 

Expenses  of  getting  the  ship  off  the  sands,  56 

Pilotage  and  port  duties  going  in  and  out  |  jqq 

of  the  harbor,  commissions,  &c.,       -       ) 

Expenses  in  port,        -----  25 

Adjusting  the  average,         -         -         -         -  4 

Postage,    -         -         -         -         -         -         -     1^ 

Total  loss,  $1186 


GENERAL   AVERAGE.  297 

ARTICLES    TO    CONTRIBUTE. 

Goods  of  A  cast  overboard,     -         -         -         -         -$500 

Value  of  B's  goods  at  N.  O.,  deducting  freight,  &c.,  1000 

"     of  C's             "             «             «             «  500 

"     of  D's             «             «             «             «  2000 

"     of  E's             «             «  •           «             «  5000 

Value  of  the  ship, 2000 

Freight,  after  deducting  one-third,    -         -         -         -  800 

S1L800 

Then,  total  value  :  total  loss  :  :  100  :  per  cent  of  loss. 
$11800    :     1180       ::  100  :  10; 

hence,  each  loses  10  per  cent  on  the  value  of  his  interest  in  the 
cargo,  ship,  or  freight.  Therefore,  A  loses  850 ;  B,  $100 ;  C, 
$50  ;  D,  $200  ;  E,  $500  ;  the  owners  of  the  ship,  $280— in  all 
$1180.  Upon  this  calculation  the  owners  are  to  lose  $280; 
but  they  aix3  to  receive  their  disbursements  from  the  contribu- 
tion, viz.,  freight  on  goods  thrown  overboard,  $100 ;  damages 
to  ship,  $200  ;  various  disbursements  in  expenses,  $180 ;  total, 
$480 ;  and  deducting  the  amount  of  contribution,  they  wiU 
actually  receive  $200.  Hence,  the  account  will  stand  : 
The  owners  are  to  receive  .  -  -  _  _  $200 
A  loses  8500,  and  is  to  contribute  $50  ;  hence,  he 


he-) 

y  450 

receives         -         -         -         -         -         "-        "   ) 

B  loses  $200,  and  is  to  contribute  $100  ;  hence,  he  "> 

'       y  100 

receives     -------        t 


Total  to  be  received,  .         -         -         $750 

C  $  50 

C,  D,  and  E,  have  lost  nothing,  and  are  to  pay        -l  D     200 

E     500 


Total  actually  paid,  -         -         -         -       $750  ; 


297.  How  is  the  freight  valued  1  How  much  is  charged  on  account  o( 
the  seamen's  wages  1  How  is  the  cargo  valued  1  Does  the  part  lost  beaj 
its  part  of  the  loss  1  How  is  the  ship  valued  1  When  parts  of  the  ship 
are  lost,  how  arc  they  compensated  for  1  How  do  you  explain  the  example ' 

5 


298  TONNAGE    OF    VESSELS. 

60  that  the  total  to  be  paid  is  just  equal  to  the  total  loss,  as  it 
should  be,  and  A  and  B  get  their  remaining  and  injured  goods, 
and  the  three  others  get  theirs  in  a  perfect  state,  after  paying 
their  ratable  proportion  of  the  loss. 


TONNAGE    OF   VESSELS. 

298.  There  are  certain  custom  house  charges  on  vessels, 
which  are  made  according  to  their  tonnasre.  The  tonage  of  a 
vessel  is  the  number  oi,  tons  weight  she  will  carry,  and  this  is 
determined  by  measurement. 

[From  the  "  Digest,"  by  Andrew  A.  Jones,  of  the  N.  Y.  Custom  House]. 

Custom  house  charges  on  all  ships  or  vessels  entering  from  any  foreign 

port  or  place. 

Ships  or  vessels  of  the  United  States,  having  three-fourths  of 

the  crew  and  all  the  officers  American  citizens,  per  ton,  SO, 06 

Ships  or  vessels  of  nations  entitled   by  treaty  to  enter  at  the 

same  rate  as  American  vessels,  .  .  -  .  .  ^06 
Ships  or  vessels  of  the  United  States  not  having  three-fourths 

the  crew  as  above,  -         -         -         -        -         -         -         -,50 

On  foreign  ships  or  vessels  other  than  those  entitled  by  treaty,  .50 
Additional    tonnage   on   foreign   vessels,    denominated   light 

money, ..-.        ,50 

Licensed  coasters  are  also  liable  once  in  each  year  to  a  duty  of  50 
cents  per  ton,  being  engaged  in  a  trade  from  a  port  in  one  slate  to  a 
port  in  another  state,  other  than  an  adjoining  state,  unless  the  officers 
and  three-fourths  of  the  crew  are  American  citizens  :  to  ascertain 
•which,  the  crew's  are  always  liable  to  an  examination  by  an  officer. 

A  foreign  vessel  is  not  permitted  to  carry  on  the  coasting  trade ; 
but  having  arrived  from  a  foreign  port  with  a  cargo  consigned  tc 
more  than  one  port  of  the  United  States,  she  may  proceed  coastwise 
with  a  certified  manifest  until  her  voyage  is  completed. 


298.  What  is  the  tonnage  of  a  vessel  ]  What  are  the  custom  house 
charges  on  the  dilTercnt  classes  of  vessels  trading  with  foreign  countries  f 
To  what  charges  arc  coasters  subject  1 


TONNAGE    OF   VESSELS.  299 

299.  The  government  estimate  the  tonnage  according  to  one 
rule,  while  the  ship  carpenter  wlio  builds  the  vessel  uses  another. 

Government  Rule. — I.  Measure,  in  feet,  above  the  vpper 
deck  the  length  of  the  vessel,  from  the  fore  part  of  the  main  stem 
to  the  after  part  of  the  stern-post.  Then  measure  tlte  breadth 
taken  at  the  ividest  part  above  the  main  ivale  on  the  outside,  and 
the  depth  from  the  under  side  of  the  deck-plank  to  the  ceiling  in 
the  hold. 

11.  From  tlte  length  take  three-fifths  of  the  breadth  and  mul- 
tiply the  remainder  by  the  breadth  a?id  depth,  and  the  product 
divided  by  95  will  give  the  tonnage  of  a  single  decker  ;  and  the 
same  for  a  double  decker,  by  merely  making  the  depth  equal  to 
half  tlte  breadth. 

Carpenters'  Rule. — Multiply  together  the  length  of  the 
keel,  the  breadth  of  the  main  beam,  and  the  depth  of  the  hold, 
and  the  product  divided  by  do  will  be  the  carpenters^  tonnage  for 
a  single  decker  ;  and  for  a  double  decker,  deduct  from  the  depth 
of  the  hold  half  the  distance  betiveen  decks. 

EXAMPLES. 

1.  What  is  the  government  tonnage  of  a  single  decker,  whose 
length  is  75  feet,  breadth  20  feet,  and  depth  17  feet  ? 

2.  What  is  the  carpenters'  tonnage  of  a  single  decker,  the 
length  of  whose  keel  is  90  feet,  breadth  22  feet  7  inches,  and 
depth  20  feet  6  inches  ? 

3.  What  is  the  carpenters'  tonnage  of  a  steamship,  double 
decker,  length  154  feet,  breadth  30  feet  8  inches,  and  depth, 
after  deducting  half  between  decks,  14  feet  8  inches  ? 

4.  What  is  the  government  tonnage  of  a  double  decker,  the 
length  being  103  feet,  breadth  25  feet  6  inches  ? 

5.  What  is  the  carpenters'  tonnage  of  a  double  decker,  its 
length  125  feet,  breadth  25  feet  6  inches,  depth  of  hold  34  feet, 
and  distance  between  decks  8  feet  ? 


299.  What  is  the  government  rule  for  finding  the  tonnage  1     What  the 
fchipbuilders'  rule  1 


800  INVOLUTION. 


INVOLUTION. 

300.  A  POWER  is  the  product  of  two  or  more  equal  factors. 
Either  of  the  equal  factors  is  called  the  root  of  the  power. 

The  root,  in  Algebra,  is  Q.dl\eAi\\Q  first  poiver. 
The  second lyower  is  the  j^roduct  of  the  root  by  itself: 
The  third  power  is  the  product  when  the  root  is  taken  3  times 
as  a  factor  : 

The  fourth  poioer,  when  it  is  taken  4  times  : 
The  fifth  power,  when  it  is  taken  5  times,  &;c. 

301.  The  number  denoting  how  many  times  the  root  is  taken 
as  a  factor,  is  called  the  exponent  of  the  power.  It  is  written 
a  little  at  the  right  and  over  the  root :  thus,  if  the  equal  factor 
or  root  is  3, 

3  =      3  the  root  or  base. 

3-  =  3  X  3  =      9  the  2d  power  of  3. 

33  =  3    x3x3i^     27  the  3d  power  of  3. 

3*  =  3x3    x3x3=    81  the  4th  power  of  3. 

35  =  3    X  3  x  3    X  3  X  3  =  243  the  5th  power  of  3. 

Involution  t's /^e  operation  of  finding  the  powers  of  numhers. 

Note. — 1.  There  are  three  things  connected  with  every  power  : 
1st,  The  root;  2d,  The  exponent :  and  3d,  The  .power  or  result  of  Iho 
multiplication. 

2.  In  finding  any  power,  one  multipliccation  gives  the  2nd  power: 
hence,  iha  number  of  multiplications  is  1  less  than  the  exponent. 

Rule. — Multiply  the  mimber  i?tto  itself  as  many  times  less  1 
as  there  are  units  in  the  exponent,  and  the  last  'product  will  be 
the  power. 

300.  What  is  a  power  1  What  is  the  root  of  a  power  1  What  is  the 
first  power !     Wliat  is  the  second  power  ]     The  third  power  ? 

301.  What  is  the  exponent  of  the  power  ?  How  is  it  written  1  What 
is  Involution  ;  How  many  things  are  connected  with  every  power  \  How 
do  you  find  the  power  of  a  number  ? 


EVOLUTION. 


'601 


EXAMPLES. 

Find  the  power  of  the  following  numbers  : 


1.  The  square  of  4? 

2.  The  square  of  15? 

3.  Tlie  square  of  26  ? 

4.  The  square  of  142  ? 

5.  The  square  of  4G3  ? 
G.  Tlie  square  of  1340? 

7.  The  square  of  24. G  ? 

8.  The  square  of  .526  ? 

9.  The  square  of  3.125  ? 

10.  The  square  of  .0524  ? 

11.  The  square  of  246.25  ? 

12.  The  square  of  -I  ? 

13.  The  square  of  |^? 

14.  The  square  of  -g-? 

15.  Tlie  square  of  }|-? 

16.  The  square  of  -| j  ? 

17.  The  square  of  yjy  ? 

18.  The  square  of  24  ? 

19.  The  square  of  7|-  ? 

20.  The  square  of  15-j\  ? 

21.  The  square  of  225^%? 


22.  The  cube  of  6  ? 

23.  The  cube  of  24  ? 

24.  The  cube  of  72  ? 

25.  The  cube  of  125  ? 

26.  The  cube  of  136? 

27.  The  4th  power  of  12  ? 

28.  The  5th  power  of  9  ? 

29.  The  value  of  (4.25)^? 

30.  The  value  of  (1.8)'? 

31.  The  value  of  (32.4)3? 

32.  The  valu«  of  (.45)5  p 

33.  The  value  of  (If)^? 

34.  The  cube  of  (|)  ? 

35.  The  4th  power  of  |  ? 

36.  The  cube  of  14f? 

37.  The  value  of  (21)^? 

38.  The  value  of  (|f)^? 

39.  The  value  of  (24|)3  ? 

40.  The  value  of  (.25)6  ? 

41.  The  value  of  (142.5)3? 

42.  The  value  of  (3.205)-  ? 


EVOLUTION. 

302.  Evolution  is  the  operation  of  finding  the  root  when 
know  the  power. 

The  Square  Root  of  a  number  is  the  factor  which  multi- 
plied by  itself  07ice  will  produce  the  number. 

The  Cube  Eoot  of  a  number  is  the  factor  which  multiplied 
by  itself  taice  will  produce  the  number. 


302.  What  is  Evolution!  What  is  the  square  root  of  a  number  7  \^'hat 
IS  the  cube  root  of  a  number  1  How  do  you  denote  the  square  root  of  a 
number !     How  the  cube  rooti 

14 


302  EXTRACTION    OF   THE   SQUARE   ROOT. 

Thus,  8  is  the  square  root  of  64,  because  8  X  8  =  64 ;  and 
3  is  the  cube  root  of  27,  because  3  X  3  x  3  =  27. 

The  sign  J  is  called  the  radical  sign.  "When  placed  before 
a  number  it  denotes  that  its  square  root  is  to  be  extracted. 
Thus,    yST  =  6. 

"VVe  denote  the  cube  root  by  the  same  sign  with  3  written 
over  it :  thus,  ^/^,  denotes  the  cube  root  of  27,  which  is  equal 
to  3.  The  small  figure  3,  placed  over  the  radical,  is  called  the 
index  of  the  root. 

EXTRACTION  OF  THE  SQUARE  ROOT. 

303.  The  square  ro  t  of  a  number  is  one  of  the  two  equal 
factors  of  that  number.  To  extract  the  square  root  is  to  find 
this  factor.     The  first  ten  numbers  and  their  squares  are 

1,      2,      3,       4,        5,        G,       7,       8,       9,        10. 
1,      4,      9,      16,      25,      36,     49,     64,     81,      100. 
The  numbers  in  the  first  line  are  tlie  square  roots  of  those  in 
the  second.     The  numbers  1,  4,  9,  16,  25,  36,  &c.,  having  two 
exact  equnl  factors,  are  called  perfect  squares. 

A  perfect  square  is  a  number  which  has  two  exact  equal  factors. 
A  perfect  square  is  a  number  which  has  two  exact  factors. 

Note. — The  square  root  of  a  number  less  than  100  will  be  less 
than  10,  while  Ihc  square  root  of  a  number  greater  than  100  i\ill  be 
greater  than  10. 

304,  To  find  the  square  root  of  a  nunxber. 

1.  "What  is  the  square  of  36  =  3  tens  +  6  units  ? 

Analysis. — The  square  of  36  is  found  by- 
taking  3fi  thirty-six  times;  and  this  is  done 
by  first  taking  it  (5  units  times  and  then  3  tens 
times,  and  adding  the  products.  3G  taken 
6  units  times,  gives  3  x  G  +  6^ :  and  taken 
3  tens  times  gives  3=  +  3  X  6.  and  their  sum  3"  +  2(3  X  fi)  +  G" 
is  3*  +  2  (3  X  G)  +  6" :  that  is, 

303.  What  is  the  square  root  of  a  number  \  What  is  a  perfect  square'? 
How  many  perfect  squares  arc  there  between  1  and  100  ]  How  do  the 
roots  of  numbers  less  than  100  compare  with  10  ! 

;J0'1.  Into  wliat  parts  may  every  inmibcT  be  decomposed  !  Wlicn  so 
J-ffiMiiiiuscd   what  is  its  square  equal  to  ! 


OPERATION. 

3  +  6 

3  +  6 

3 

X  6  +  6= 

3^+3 

X  6 

EXTRACTION    OF    THE    SQUAKE    ROUT. 


303 


F 

30 

I 

30 

6 

6 

6 

TT 

180 

3G 

30 

E 

O 

o 

n 

80 
30 

900 

C8 

30 
6 

180 

D 


K 


M 


A 


30 


B 


T/ie  square  of  a  nicmher  is  equal  to  the  square  of  the  tens,  vlus 
twice  the  'product  of  the  tens  hy  the  units,  jjIus  the  square  of  ''.'C 
units. 

The  same  may  be  shown  by  a  diagram : 

Let  the  line  AB  re- 
present the  3  tens  or  30, 
and  BC  the  six  units. 

Let  AD  be  a  square 
on  AC,  and  AE  a  square 
on  the  ten's  line  AB. 

Then  ED  will  be  a 
square  on  the  unit  line 
6,  and  tlie  rectangle  EF 
■will  be  the  product  of 
HE,  which  is  equal  to 
the  ten's  line,  by  IE, 
which  is  equal  to  the 
unit   line.       Also,    the 

rectangle  BK  will  be  the  product  of  EB.  which  is  equal  to  the  ten's 
line,  by  the  unit  line,  BC.  But  the  whole  square  on  AC  is  made  up 
of  the  .'^quare  AE,  the  two  rectangles  FE  and  EC  and  the  square  ED  . 
hence,  it  is  equal  to 

The  square  of  the  tens  plus  t/rice  the  product  of  the  tens  hy  the 
units,  plus  the  square  of  the  units. 

1.  Let  it  nov/  be  required  to  extract  the  square  root  of  2025. 

Analysis. — Since  the  number  contains  more  than  two  places  of 
figures,  its  root  will  contain  tens  and  units.  But  as  the  square  of  one 
ten  is  one  hundred,  it  follows  that  the  square  of  the  tens  of  the 
required  root  must  be  found  in  the  two  figures  on  the  left  of  25. 
Hence,  beginning  at  the  right,  we  point  off  the  number  into  periods 
of  two  figures  eacli. 

We  next  find  the  greatest  square  contained  in 
20  tens,  which  is  4  tens  or  40.  We  then  square 
4  tens  which  gives  16  hundred,  and  then  place 
16  under  the  second  period,  and  subtract ;  this 
takes  away  the  square  of  the  tens,  and  leaves 
425,  which  is  twice  the  product  of  the  tens  by  the 
units  plus  the  square  of  the  units. 


OPERATION. 

20 

2'c 

(85 

16 

8 

5)42 

5 

42 

5 

o04  EXTRACTION   OF   THE   SQUARE   ROOT. 

If  now,  we  double  the  tens  and  then  divide  the  remainder  exclu- 
sive of  the  right  hand  figure,  (since  that  figure  cannot  enter  into  the 
product  of  the  tens  by  the  units)  by  it,  the  quotient  will  be  the  units 
figure  of  the  root.  If  we  annex  this  figure  to  the  augmented  divisor, 
and  then  multiply  the  whole  divisor  tiius  increased  by  it.  the  pro- 
duct will  be  twice  the  tens  by  the  units  plus  the  square  of  the 
units  :  and  hence,  we  have  found  both  figures  of  the  root. 

This  process  may  also  be  illustrated  by  the  diagram. 

Suppose  AC  =  45.  Then,  subtracting  the  square  of  the  tens  is 
taking  away  the  square  AE,  and  leaves  the  two  rectangles  FE  and 
BK,  together  with  the  square  ED  on  the  unit  line. 

The  two  rectangles  FE  and  BK  representing  the  product  of  units 
by  tens,  can  be  expressed  by  no  figures  less  than  tens. 

If  then,  "we  divide  the  number  42,  at  the  left  of  5.  by  twice  the 
tens,  that  is,  by  twice  AB  or  BE,  the  quotient  will  be  BC  or  EK,  the 
unit  of  the  root. 

Then,  placing  BC  or  5,  in  the  root,  and  also  annexing  it  to  the 
divisor  doubled,  and  then  multiplying  the  whole  divisor  85  by  5.  wc 
obtain  the  two  rectangles  FE  and  CE,  together  with  the  square  ED 

305.  Hence,  for  the  extraction  of  the  square  root,  we  nave 
the  following 

lluLE. — I.  Separate  tlie  given  number  into  periods  of  tico 
figures  each,  hy  setting  a  dot  over  the  pfoce  of  units,  a  second 
over  the  place  of  hundreds,  and  so  on  for  each  cdternate  figure 
to  the  left. 

II.  Note  the  greatest  square  contained  in  the  j^eriod  on  the  left, 
and  place  its  root  on  the  rigid  after  the  manner  of  a  quotient  in 
division.  Subtract  the  squai-e  of  this  root  from  the  first  period, 
and  to  the  remainder  bring  down  the  second  period  for  a  divi- 
dcnd. 

III.  Double  the  root  thus  found  for  a  trial  divisor  and  place 
ii  on  the  left  of  the  dividend.  Firid  hoiv  7nang  times  the  trial 
divisor  is  contained  in  the  dividenJ,  exclusive'  of  the  right-hand 


305.  What  is  the  first  step  in  extracting  the  square  root  of  numliors  \ 
Wlr.t  is  tlic  second  1  What  is  the  third  1  Wliat  the  fourth  ?  \Vlial  tho 
iW'ili  \     (Ilvo  tiic  entire  rule] 


EXTKACTION    OF   Tnii   SQUARE   BOOT.  305 

figure,  and  place  the  quotient  in  the  root  and  also  annex  it  to  the 
divisor. 

IV.  Midtiply  the  divisor  thus  increased,  hi/  the  last  figure  of 
the  root;  subtract  the  product  from  the  dividend,  and  to  the 
remainder  bring  down  the  next  period  for  a  new  dividend. 

V.  Boxdjle  the  whole  root  thus  found,  for  a  new  trial  divisor, 
and  contimie  the  operation  as  before,  until  all  the  periods  are 
brought  down. 

EXAMPLES. 

1.  What  is  the  square  root  of  425104? 

Analysis. — We  first  place  a  dot  over  the  operation. 

4,  making  the  right-hand   period  04.     We  42  51   04(652 

then  put  a  dot  over  the  1.  and  also  over  the  36 

2,  making  three  periods.  125)651 

The  greatest  perfect  square  in  42  is  36,  625 

the  root  of  which   is   6.     Placing  6  in  the         1302)2604 
root,    subtracting  its  square  from  42,   and  2604 

bringing  down  the  next  pcriod"51.  we  have 

651  for  a  dividend,  and  by  doubling  the  root  we  have  12  for  a  trial 
divisor.  Now,  12  is  contained  in  Go.,  5  times.  Place  5  both  in  the 
root  and  in  the  divisor;  then  multiply  125  by  5  ;  subtract  the  pro- 
duct and  bring  down  the  next  period. 

We  must  now  double  the  whole  root  65  for  a  new  trial  divisor;  or 
we  may  take  the  first  divisor  after  having  doubled  the  last  figure  5; 
then  dividing,  we  obtain  2.  the  third  figure  of  the  root. 

Notes. — 1.  Tlie  left-hand  period  may  contain  but  one  figure  ;  each 
of  the  others  will  contain  two. 

2  If  any  trial  divis^or  is  greater  than  its  dividend,  the  correspond- 
ing quotient  figure  will  be  a  cipher. 

3.  If  the  product  of  the  divisor  by  any  figure  of  the  root  exceeds 
the  corresponding  dividend,  the  quotient  figure  is  too  large  and  must 
be  diminished. 

4.  There  will  be  as  many  figures  in  the  root  as  there  are  periods 
in  the  given  number. 

5.  If  the   given  number   is  not  a  perfect  square  there  will  be  a 

remainder  after  all  the  periods   are  brought  down.      In  this  casoj 

periods  of  ciphers  may  be   annexed,  forming   new  periods,  each  of 

which  will  give  one  decimal  place  in  the  root. 

14 


306 


EXTEACTION    OF    THE    SQUARE    KOOT. 


2.  What  is  the  square  root  of  758692  ? 


OPERATION. 

75  86  92(871.029  +. 
64 


Note. — After  using  all  the 
periods  of  the  given  number, 
we  annex  periods  of  decimal 
ciphers,  each  of  which  gives 
one  decimal  place  in  the  quo- 
tient. 


167)11 
11 


86 
69 


1741)17  92 
17  41 


174202)510000 
348404 


1742049)16159600 
15678441 

481159   Rem, 


306.   To  extract  the  square  root  of  a  fraction 

1.  "What  is  the  square  root  of  .6  ? 

Note. — We  first  annex  one  cipher  to  make 
even  decimal  places;  for.  one  decimal  multi- 
plied by  itself  will  give  two  places  in  the 
product.  We  then  extract  the  root  of  the 
first  period,  and  to  the  remainder  annex  a 
decimal  period,  and  so  on,  till  we  have  found 
a  sufficient  number  of  decimal  places. 


2.  What  is  the  square  root  of  if  ? 

Note. — The  square  root  of  a  fraction 
is  equal  to  the  square  root  of  the  nume- 
rator divided  by  the  square  root  of  the 
denominator. 

3.  What  is  the  square  root  of  |^  ? 

NoTK. — When  the  terms  are  not  per- 
fect square.",  reduce  the  common  fraction 
to  a  decimal,  and  then  extract  the  square 
root  of  the  decimal. 


OPERATION. 

.60(.774  + 

49 

147)1100 

1029 

1544)7100 

6176 

924  rem. 

OPERATION. 

fxS.   Vl  6    _    4 

2^    -V2^-** 


OPERATION. 

f  =  .75  ; 
yf'=  y/^  =  .8545  •+• 


SOf).  How  do  you  extract  the  square  root  of  a  decimal  number  \     How 
of  a  coiiirnon  fraction  \ 


EXTKACTION  OF  THE  SQUARE  KOOT. 


307 


Rule. — I.  If  ike  fraction  is  a  decimal,  point  off  the  periods 
from  the  decimal  point  to  the  rir/lU,  annexing  cipliers  if  neces- 
sary, so  that  each  period  shall  contain  two  places,  and  then  ex- 
tract the  root  as  in  integral  numbers. 

11.  If  the  fraction  is  a  common  fraction,  and  its  terms  perfect 
squares,  extract  the  square  root  of  the  numerator  and  denomina- 
tor separately  ;  if  they  are  not  perfect  squares,  reduce  the  frac- 
tion to  a  decimal,  and  then  extract  the  square  root  of  the  result. 

EXAMPLES. 

What  are  the  square  roots  of  the  following  numbers  ? 


1.  Square  root  of  49  ? 

2.  Square  root  of  144  ? 

3.  Square  root  of  225  ? 

4.  Square  root  of  2304  ? 

5.  Square  root  of  ff  ? 

6.  Square  root  of  o^^^  ? 

7.  Square  root  of  .0196? 

8.  Square  root  of  6.25  .'' 

9.  Square  root  of  278.89? 

10.  Square  root  of  6275025  ? 

11.  Square  root  of  7994  ? 

12.  Square  root  of  .205209  ? 

13.  Square  root  of  |  ? 

14.  Square  root  of  ^|? 

15.  Square  root  of  Jg-  ? 


16.  Square  root  of  .60794  ? 

17.  Value  of  ^-022201  ? 

18.  Value  of  -^^25.1001  ? 

19.  Value  of  -v/196.425  ? 

20.  Value  of  VTS  ? 


21.  Value  of  VffH  ? 

22.  Value  of  ^  ? 


23.  Value  of 


25  • 


24.  Value  of  yT35  ? 


25.  Value  of  -^19000  ? 

26.  Value  of  yT784? 

27.  Square  root  of  5647.5225  ? 

28.  Square  root  of  160048.003  6! 


APPLICATIONS   IN    SQUARE    ROOT. 

307.  A  triangle  is  a  plain  figure  Avhich  has  three  sides  and 
three  angles. 

If  a  straifjht  line  meets  another  straio;ht  line, 

C5  0  7 

making  the  adjacent  angles  equal,  each  is  called 
a  right  angle ;  and  the  lines  arc  said  to  be  per- 
pendicular  to  each  other. 


'.507.  Wliat  is  a  triangle  ?     What  is  a  iglit  angle 


308 


EXTKACTION  OF  THE  SQUAKE  KOOT. 


308.  A  right  angled  triangle  is  one  which 
has  one  right  angle.  In  the  right  angled  tri- 
angle ABC,  the  side  AC  opposite  the  right 
angle  B,  is  called  the  hypotheniise  ;  the  side 
AB  the  base  ;  and  the  side  BC  the  perpen- 
dicular. 

309.  In  a  right  angled  triangle  the  square  described  on  the 
hypothenuse  is  equal  to  the  sum  of  the  squares  described  on  the 
other  two  sides. 

Thus,  if  ACB  be  a  right 
angled  triangle,  right  an- 
gled at  C,  then  will  the 
large  square,  D,  described 
in  the  hypothenuse  AB,  be 
equal  to  the  sum  of  the 
squares  F  and  E  described 
on  the  sides  AC  and  CB. 
This  is  called  the  carpen- 
ter's theorem.  By  count- 
ing the  small  squares  in  the 
large  square  D,  you  will 
find  their  number  equal 
to   that    contained    in    the 

small  squares  F  and  E.  In  this  triangle  the  hypothenuse 
AB  =  5,  AC  —  4,  and  CB  =  3.  Any  numbers  having  the 
same  ratio,  as  5,  4  and  3,  such  as  10,  8  and  6 ;  20,  16  and  12, 
&c.,  will  represent  the  sides  of  a  right  angled  triangle. 

310.  When  the  base  and  perpendicular  are  known,  to  find  the 
hrjuotlienusc. 


D 

— 

308.  What  is  a  ri^ht  angled  triangle  ?    Which  side  is  the  hypothenuse] 

309.  In  a  right  angled  triangle,  what  is  the  square  on  the  hyjjolhenuso 
fq\ial  to  1 

310.  How  do  you  find  the  hypothenuse  when  you  know  the  bise  and 
perpendicular  1 


EXTRACTION    OF   THE   SQUARE    KOOT.  309 

1.  "Wishing  to  know  the  distance  from  A 
to  the  top  of  11  tower,  I  measured  the  lieight 
of  the  tower  and  found  it  to  be  40  feet ;  also 
the  distance  from  A  to  B,  and  found  it  30 
feet :  Avhat  was  the  distance  from  A  to  C  ? 
AB  =  30 ;  AB-  =  30^  =    900 
BC  =  40 ;  BC2  =  402  ^  iGOO 
AC-  =  AB2  +  BC2  =  900  +1600 
AC  =  y'2500  =:  50  feet. 

Rule. — Square  the  base  and  square  the  2Jei-pendicular,  add 
the  results,  ami  then  extract  the  square  root  of  their  su)7i. 

311.  To  Jiiid  one  side  ivhen  we  know  the  hypothenuse  and  the 
other  side. 

1.  The  length  of  a  hidder  which  will  reach  from  the  middle 
of  a  street  80  feet  wide  to  the  eaves  of  a  house,  is  50  feet : 
■what  is  the  height  of  the  house  ? 

Analysis. — Since  the  square  of  the  length  of  the  ladder  is  equal 
to  the  sum  of  the  squares  of  half  the  width  of  the  s-trect  and  the 
height  of  the  house,  the  square  of  the  length  of  the  ladder  diminished 
by  the  square  of  half  the  width  of  the  street  will  be  equal  to  the 
square  of  the  heighl  of  the  house :  hence, 

EuLE. — S'juare  the  hypothenuse  and  the  hnoion  side,  and  take 
the  difference  ;  the  square  root  of  the  difference  will  be  the  other 
side. 

EXAMPLES. 

1.  A  general  having  an  army  of  117649  men,  Avished  to  form 
them  into  a  square  :  how  many  should  he  place  on  each  front  ? 

2.  In  a  square  piece  of  pavement  there  are  48841  stones,  of 
equal  size,  one  foot  square  :  what  is  the  length  of  one  side  of 
the  pavement  ? 

3.  In  the  centre  of  a  square  garden,  there  is  an  artificial 
circular  pond  covering  an  area  of  810  square  feet,  which  is  y^ 


311.   When  j'ou  know  the  hypothenuse  and  one  side,  how  do  you  find 
lh(.-  other  i;ide  ! 


310        EXTK ACTION  OF  THE  SQUARE  EOOT. 


of  the  whole  garden :  how  many  rods-  of  fence  will  enclose  tho 
garden  ? 

4.  Let  it  be  required  to  lay  out  67  J.  2R.  of  land  in  the  form 
of  a  rectangle,  the  longer  side  of  which  is  to  be  three  times  as 
great  as  the  less  :  what  is  its  length  and  width  ? 

5.  A  farmer  wishes  to  set  out  an  orchard  of  3200  dwarf  pear 
trees.  He  has  a  field  which  is  twice  as  long  as  it  is  wide  which 
he  appropriates  to  this  purpose,  setting  the  trees  12  feet  apart- 
each  way :  how  many  trees  will  there  be  in  a  row,  each  way, 
and  how  much  land  will  they  occupy  ? 

G.  There  is  a  wall  45  feet-  high  built  upon  the  bank  of  a 
stream  GO  feet  wide  :  how  long  must  a  ladder  be  that  will  reach 
from  the  outside  of  the  stream  to  the  top  of  the  wall  ? 

7.  A  boy  having  lodged  his  kite  in  the  top  of  a  tree,  finds 
that  by  letting  out  the  whole  length  of  his  line,  which  he  knows 
to  be  225  feet,  it  will  reach  the  ground  180  feet  from  the  loot 
of  the  tree  :  what  is  the  height  of  the  tree  ? 

8.  There  are  two  buildings  standing  on  opposite  sides  of  the 
street,  one  39  feet,  and  the  other  49  feet  from  the  ground  to  the 
eaves.  The  foot  of  a  ladder  Q)b  feet  long  rests  upon  the  ground 
at  a  point  between  them,  from  which  it  will  touch  the  eaves  of 
either  building :  what  is  the  width  of  the  street  ? 

9.  A  tree  120  feet  higli  was  broken  off  in  a  storm,  the  top 
striking  40  feet  from  the  roots,  and  the  broken  end  resting  upon 
the  stump  :  allowing  the  ground  to  be  a  horizontal  plane,  what 
was  the  height  of  tlie  part  standing  ? 

10.  What  will  be  the  distance  from  corner  to  corner,  tln-ough 
the  centre  of  a  cube,  whose  dimensions  are  5  feet  on  a  side  ? 

1 1 .  Two  vessels  start  from  the  same  point,  one  sails  due 
north  at  the  rate  of  10  miles  an  liour,  the  other  due  west  at  the 
rate  of  14  miles  an  hour:  how  far  apart  will  they  be  at  the 
end  of  2  days,  supposing  the  surface  of  the  earth  to  be  a  plane? 

12.  How  much  more  will  it  cost  to  fence  10  acres  of  land,  in 
the  form  of  a  rectangle,  the  length  of  which  is  four  times  its 
breadth,  llian  if  it  were  in  the  form  of  a  square,  the  cost  of  thu 
fence  being  $2.50  a  rod  ? 


CUBE    KOOT.  311 

13.  "^Yhat  is  the  diameter  of  a  cylindrical  resei'voir  contain- 
ins:  9  limes  as  much  water  as  one  25  feet  in  diameter,  the 
heiijhts  beino;  the  same  ?* 

14.  If  a  cylindrical  cistern  8  feet  in  diameter  will  hold  120 
barrels,  what  must  be  the  diameter  of  a  cistern  of  the  same  depth 
to  hold  1500  barrels? 

15.  If  a  pipe  3  inches  in  diameter  will  discharge  400  gallons 
in  3  minutes,  what  must  be  the  diameter  of  a  pipe  that  will 
discharo-e  IGOO  gallons  in  the  same  time? 

IG.  "What  length  of  rope  must  be  attached  to  a  halter  4  feet 
long  that  a  horse  may  feed  over  21  acres  of  ground  ? 

17.  Three  men  bou2;ht  a  si-rindstone,  Avhich  was  four  feet  in 
diameter :  how  much  must  each  grind  off  to  use  up  his  share 
of  the  stone  ? 


CUBE    ROOT. 

312.  The  Cube  Root  of  a  number  is  one  of  three  equal 
factors  of  the  number. 

To  extract  the  cube  root  of  a  number  is  to  find  a  factor  whicli 
multiplied  into  itself  twice,  will  produce  the  given  number. 

Thus,  2  is  the  cube  root  of  8  ;  for,  2  X  2  X  2  =:  8 :  and  3  b 
the  cube  root  of  27 ;  for,  3  x  3  x  3  =  27. 

1,         2,         3,         4,         5,         6,         7,         8,         9, 
1  8         27        64      125       216     343      512      729 

The  numbers  in  the  first  line  are  the  cube  roots  of  the  cor- 
responding numbers  of  the  second.  The  numbers  of  the  second 
line  are  called  perfect  cubes.     A  number  is  a  iX'rfccl  cube  when 

313.  What  is  the  cube  root  of  a  number  \  "When  is  a  number  a  perfect 
cube  1     How  many  perfect  cubes  are  there  between  1  and  1000  ] 

*  Note  — If  two  volumes  have  the  same  altitude,  their  contents  will  be 
to  each  other  in  the  same  proportion  as  their  bases  ;  and  if  the  bases  are 
similar  figures  (that  is,  of  like  form,)  they  will  be  to  each  other  as  the 
squares  of  iheir  diameters,  or  other  like  dimensions. 


312  CUBE    ROOT, 

it  has  three  exact  equal  factors.     By  examining  the  numbers 
in  the  two  lines  we  see, 

1st.  That  the  cube  of  units  cannot  give  a  liigher  order  than 
hundreds. 

2d.  That  since  the  cube  of  one  ten  (10)  is  1000  and  the  cube 
of  9  tens  (*J0),  729000,  tJie  cube  of  tens  ivill  not  give  a  lower 
denomination  than  thousands,  nor  a  higher  denomination  than 
hundreds  of  thousands. 

Hence,  if  a  number  contains  more  than  three  figures,  its  cube 
root  will  contain  more  than  one  ;  if  it  contains  more  than  sis, 
its  I'oot  Avill  contain  more  than  two,  and  so  on  ;  every  additional 
three  figures  giving  one  additional  figure  in  the  root,  and  the 
figures  Avhich  remain  at  the  left  hand,  although  less  than  three, 
will  also  give  a  figure  in  the  root.  This  law  explains  the 
reason  for  pointing  off  into  periods  of  three  figures  each. 

313.  Let  us  now  see  how  the  cube  of  any  number,  as  10,  is 
formed.  Sixteen  is  composed  of  1  ten  and  6  units,  and  may  be 
■written,  10  +  0.  To  find  the  cube  of  10  =  10  +  6,  we  must 
multiply  the  number  by  itself  twice. 

To  do  this  we  place  the  number  thus, 

product  by  the  units,  _         -         - 

product  by  the  tens,  -         -         . 

Square  of  16 
Multiply  again  by  16, 
product  by  the  units,  -         -         -         600+    720  +  216 

product  by  the  tens,  -         -         -  1000  +  1200  +    300 

Cube  of  16  -         -         -         -  1000  +  1800  +  1080  +  216 

1.  By  examining  the  parts  of  this  number,  it  is  seen  that  the 
first  part  1000  is  the  chIjc  of  the  tens  ;  that  is, 

10  X  10  X  10  =  1000. 


16  =  10  + 
10  + 
60  + 
100  +   60 

6 
6 

3G 

100  +  120  + 
10  + 

36 
6 

yi3.  Of  how  many  parts  is  the  cube  of  a  number  composed  1     \Vhal 
arc  I  hey  ! 


CUBE   ROOT.  313 

2.  The  second  part  1800  is  three  times  the  square  of  the  tens 
multiplied  by  the  units  ;  that  is, 

3  X  (10)2  x6  =  3xl00xG  =  1800. 

3.  The  thii-d  part  1080  is  three  times  the  square  of  the  units 
multiplied  by  the  tens  ;  that  is, 

3  X  62  X  10  =  3  X  36  X  10  =  1080. 

4.  The  fourth  part  is  the  cube  of  the  units ;  that  is, 

63  =  6x6x6  =  216. 
1.  "What  is  the  cube  root  of  the  number  4096  ? 

Analysis. — Since  the  number  contains  more  than  three  figures,  we 
know  that  the  root  will  contain  at 
least  units  and  tens.  operation. 

Separating  the  three  right-hand  '  4  096(16 

figures  from  the  4,  we  know  that  1 

the  cube  of  the  tens  will  be  found  1*  X  3  =  3)3~0     (9-8-7-6 

in  the  4  ;  and  1  is  the  greatest  cube  16'  =  4  096. 

in  4. 

Hence,  we  place  the  root  1  on  the  right,  and  this  is  the  tens  of  the 
required  root.  We  then  cube  1  and  subtract  the  result  from  4,  and 
to  the  remainder  we  bring  down  the  first  figure  0  of  the  next  period. 

We  have  seen  that  the  second  part  of  the  cube  of  16,  viz.,  1800, 
is  three  times  the  square  of  the  tens  multiplied  by  the  units;  and 
hence,  it  can  have  no  significant  figure  of  a  less  denomination  than 
hundreds.  It  must,  therefore,  make  up  a  part  of  the  30  hundreds 
above.  But  this  30  hundreds  also  contains  all  the  hundreds  which 
come  from  the  3d  and  4th  parts  of  the  cube  of  16.  If  it  Mere  not  so, 
the  30  hundreds,  divided  by  three  times  the  square  of  the  tens,  would 
give  the  unit  figure  exactly. 

Forming  a  divisor  of  three  times  the  square  of  the  tens,  we  find 
the  quotient  to  be  ten ;  but  this  we  know  to  be  too  large.  ^Placing  9 
in  the  root  and  cubing  19,  we  find  the  result  to  be  6859.  Then  trying 
8  we  find  the  cube  of  18  still  too  large;  but  when  we  take  G  we  find 
the  exact  number.     Hence,  the  cube  root  of  4096  is  16. 

314,  Hence,  to  find  the  cube  root  of  a  number: 

Rule. — T.  Separate  the  given  number  into  pteriods  of  three 
fiyures  each,  by  placing  a  dot  over  the  place  of  units,  a  second 


81-i  OUBK    KOOT. 

over  the  ])lace  of  thovsa7ids,  and  so  on  over  each  third  figure  ta 
the  left:  the  left  hand  period  will  often  contain  less  than  three 
places  of  figures. 

II.  JV^ote  the  greatest  perfect  cube  in  the  first  period,  and  set 
its  root  on  the  right,  after  the  manner  of  a  quotient  in  division. 
Subtract  the  cube  of  this  number  fro )n  the  first  period,  and  to  the 
remainder  bring  down  the  first  figure  of  the  next  jieriod  for  a 
dividend. 

III.  Take  three  times  the  square  of  the  root  just  found  for  a 
trial  divisor,  and  see  how  often  it  is  contained  in  the  dividend, 
and  place  the  quotient  for  a  second  figure  of  the  root.  Then 
cube  the  figures  of  the  root  thus  found,  and  if  their  cube  be  greater 
than  the  first  tzvo  j^eriods  of  the  given  nujuber,  diminish  the  last 
figure,  but  if  it  be  less,  subtract  it  from  the  first  two  periods,  and 
to  the  remainder  bring  dotvn  the  first  figure  of  the  next  period  for 
a  new  dividend. 

IV. —  Tahe  three  times  the  square  of  the  whole  root  for  a 
second  trial  divisor,  and  find  a  third  figure  of  the  root  as  before. 
Cube  the  whole  root  thus  found  and  subtract  the  result  from  the 
first  three  jyeriods  of  the  given  numhir  when  it  is  less  than  that 
number,  but  if  it  is  greater,  diminish  the  last  figure  of  the  root, 
proceed  in  a  similar  way  for  all  the  periods. 

EXAMPLES. 

1    What  is  the  cube  root  of  2070  G875  ? 

OPERATION. 

20  796  875(275 
2'=    8 
2'  X  3  =  12)127 
First  two  periods,      ...     20  796 
(27)'  =  27  X  27  X  27  =  19  6S3 

3  X  (27)'  =  2217)1  i  138 
Tirst  three  periods,     -     -     -     20  796  875 
(275)' =  275  X  275  X  275  =  20  796  875 


314.  WTiat  is  the  rule  for  oxtracting  the  cube  root  of  a  number? 


CUBE   KOOT.  313 


Find  the  cube  roots  of  the  followinnr  numbers  : 


1.  Cube  root  of  1728? 

2.  Cube  root  of  117649? 

3.  Cube  root  of  46G56? 

4.  Cube  root  of  15069223  ? 


5.  Cube  root  of  5735339  ? 

6.  Cube  root  of  48228544  ? 

7.  Cube  root  of  84604519  ? 

8.  Cube  root  of  28991029248? 


315.  To  extract  the  cube  root  of  a  decimal  fraction  : 
Annex  ciphers  to  the  decimal,  if  necessary,  so  /hat  it  shall  con- 
sist q/  3,  6,  9,  ^c,  decimal  places .  Then  put  the  first  point  over  the 
'place  of  tlioiisaudths,  the  second  over  the  place  of  milliontlts,  and 
so  on  over  every  third  place  to  the  rigid ;  after  which  extract 
the  root  as  in  whole  numhers. 

Notes. — 1.  There  will  be  as  many  decimal  places  in  the  root  as 
there  are  periods  in  the  given  number. 

2.  Tlie  same  rule  applies  when  the  given  number  is  composed  of  a 
"whole  number  and  a  decimal. 

3.  If  in  extracting  the  root  of  a  number  there  is  a  remainder  after 
all  the  periods  have  been  brought  down,  periods  of  ciphers  may  be 
annexed  by  considering  them  as  decimals. 

EXAMPLES. 

Find  the  cube  roots  of  the  following  numbers  : 

5.  Cube  root  of  .387420489  ? 

6.  Cube  root  of  .000003375  ? 


1.  Cube  root  of  8.343  ? 

2.  Cube  root  of  1728.729? 

3.  Cube  root  of  .0125  ? 

4.  Cube  root  of  19683.46656? 


7.  Cube  root  of  .0066592? 

8.  Value  of  y8Tr7"29? 

316.  To  extract  the  cube  root  of  a  common  fraction, 

I.  Reduce  compound  fractions  to  simple  ones,  mixed  numhers 
to  improper  fractions,  and  then  reduce  the  fraction  to  its  lowest 
terms. 

314.  What  is  the  rule  for  extracting  the  cube  root  of  a  number  1 

315.  How  do  you  extract  the  cube  root  of  a  decimal  fraction  '^  How 
many  decimal  places  will  there  be  in  the  root  \  W"\\\  the  same  rule  apply 
when  there  is  a  whole  num!)er  and  a  decimal  I  If  in  extracting  the  root 
of  any  number  you  find  a  decimal,  how  do  you  proceed  T 


816  CUBE   ROOT. 

II.  Extract  the  cube  root  of  the  numerator  and  denominator 
separately,  if  they  have  exact  roots  ;  but  if  either  of  them  has  not 
an  exact  root,  reduce  (he  fraction  to  a  decimal,  and  extract  the. 
root  as  in  the  last  case. 

EXAMPLES. 

Find  the  cube  roots  of  the  following  fractions : 


1.  Cube  root  of  y"^  ? 

2.  Cube  root  of  III? 

3.  Cube  root  of  SIJ^? 

4.  Cube  root  of  911? 

5.  Cube  root  of  141? 


6.  Cube  root  of  yf^  ? 

7.  Cube  root  of  oVVrir  ? 

8.  Cube  root  of  If  ^f^? 

9.  Cube  root  of  7f  ? 
10.  Cube  root  of  56f  ? 


APPLICATIONS. 

1.  What  must  be  the  dimensions  of  a  cubical  bin,  that  its 
A'ohime  or  capacity  may  be  19683  feet? 

2.  If  a  cubical  body  contains  6859  cubic  feet,  what  is  the 
length  of  one  side  :  what  the  area  of  its  surface  ? 

3.  The  volume  of  a  globe  is  46656  cubic  inches  :  w'hat  would 
be  the  side  of  a  cube  of  equal  solidity  ? 

4.  A  person  wished  to  make  a  cubical  cistern,  which  should 
hold  150  barrels  of  water ;  what  must  be  its  depth  ? 

5.  A  farmer  constructed  a  bin  that  would  contain  1500  bush- 
els of  grain  ;  its  length  and  breadth  were  equal,  and  each  half 
the  height ;  what  were  its  dimensions  ? 

6.  AVhaL  is  the  difference  betAveen  half  a  cubic  yard,  and  a 
rube  whose  edge  is  half  a  yard  ? 

7.  7l  merchant  paid  $911,25  for  some  pieces  of  muslin.  He 
paid  as  many  cents  a  yard  as  there  were  yards  in  each  piece, 
and  there  were  as  many  pieces  as  there  were  yards  in  one 
piece  :  how  many  yards  were  there,  and  how  much  did  he  pay 
a  yard  ? 

Notes. — 1.  Bodies  are  said  to  be  similar  when  they  have  the  same 
form  and  have  their  like  parts  proportional. 

2.  It  is  proved  in  Geometry,  that  the  volumes  or  weights  of  similar 
bodies  are  to  each  other  ns  the  cubes  of  their  like  dimensions. 

3.  Those  bodies  which  arc  named  in  the  same  example  arc  sup 
posed  to  bo  similar. 


ARITHMETICAL   PROGRESSION.  317 

8.  If  a  sphere  3  feet  in  diameter  contains  14.1372  cubic  feet, 
what  are  the  contents  of  a  sphere  6  feet  in  diameter  ? 

33     :     63     ::     14.1372     :     113.0976.  Ans. 

9.  If  a  ball  2^  inches  in  diameter  weighs  8  pounds,  how  much 
will  one  of  the  same  kind  weigh,  that  is  5  inches  in  diameter  ? 

10.  What  must  be  the  size  of  a  cubical  bin,  that  will  contain 
8  times  as  much  as  one  that  is  4  feet  on  a  side  ? 

11.  How  many  globes,  6  inches  in  diameter,  will  it  require 
to  make  one  12  inches  in  diameter? 

12.  If  a  ball  of  silver,  1  unit  in  diameter,  be  worth  $8,  what 
will  be  the  value  of  one  51  units  in  diameter  ? 

13.  If  a  plate  of  silver,  6  inches  long,  3  inches  wide,  and 
i  inch  thick,  be  worth  $100,  what  will  be  the  dimensions  of  a 
similar  plate  of  the  same  metal  worth  $800  ? 

14.  If  one  man  can  dig  a  cellar  12  feet  long,  10  feet  wide, 
and  41  feet  deep,  in  3  days,  what  will  be  the  dimensions  of  a 
similar  cellar  that  requires  him  24  days  to  3ig  it,  working  at 
the  same  rate,  and  the  ground  being  of  the  same  degree  of 
hardness  ? 

15.  If  I  put  2  tons  of  hay  in  a  stack  10  feet  high,  how  high 
must  a  similar  stack  be  to  contain  16  tons  ? 

16.  Four  women  bought  a  ball  of  yarn  6  inches  in  diameter, 
and  agreed  that  each  should  take  her  share  sejiarately  from  the 
surface  of  the  ball :  how  much  of  the  diameter  must  each  wind 
off? 

ARITHMETICAL    PROGRESSION. 

317.  If  we  take  any  number,  as  2,  we  can,  by  the  continued 
addition  of  any  other  number,  as  3,  form  a  series  of  numbers : 
thus, 

2,    5,    8,    11,    14,    17,    20,    23,    Sec, 

in  which  each  number  is  formed  by  the  addition  of  3  to  the 
preceding  number. 

,S17.  What  is  an  arithmetical  progression  1  What  is  the  number  added 
or  subtracted  called  ? 


515  AKmrMETICAL   PROGRESSION. 

This  series  of  numbers  may  also  be  formed  by  subtracting  3 
continually  from  a  larger  number  :  thus, 

23,    20,    17,    14,    11,    8,    5,    2. 

An  Arithmetical  Progression  is  a  series  of  numbers  in 
which  each  is  derived  from  the  preceding  by  the  addition  or 
subtraction  of  the  same  number. 

The  number  which  is  added  or  subtracted  is  called  the  com- 
mon difference. 

318.  When  the  series  is  formed  by  the  continued  addition 
of  the  common  difference,  it  is  called  an  increasing  series ;  and 
when  it  is  formed  by  the  subtraction  of  the  common  diflference, 
it  is  called  a  decreasing  series  :  thus, 

2,     5,     8,  11,  14,  17,  20,  23,  is  an  increasing  series. 
23,  20,  17,  14,  11,     8,     5,     2,  is  a  decreasing  series. 

The  several  numbers   are  called  terms  of  the  progression 
the  first  and  last  terms  are  called  the  extremes,  and  the  interme- 
diate terms  are  called  the  tncaas. 

319.  In  every  arithmetical  progression  there  are  five  parts, 
any  three  of  wliich  being  given  or  known,  the  remaining  two 
can  be  determined.     They  are, 

1st :  The  first  term  ; 

2d :    The  last  term  ; 

3d  :    The  common  difference  ; 

4tli :  Tlie  number  of  terms  ; 

5th :  The  sum  of  all  the  terms. 


318.  When  the  common  difference  is  added,  what  is  the  series  called  1 
What  is  it  called  when  the  common  difl'crencc  is  subtracted  1  What  are 
the  several  numbers  called  1  What  are  the  first  and  last  called  1  What 
arc  the  intermediate  ones  called  ! 

319.  How  many  parts  are  there  in  every  arithmetical  progression  ^  What 
are  they!  How  many  parts  must  be  given  before  the  remaining  ones  can 
be  found  1 


AKITIIMETICAL   PROGRESSION.  319 

CASE   I. 

320.  Hating  given  the  first  term,  the  common  difference,  and 
the  nuinher  of  terms,  iojind  the  lust  term. 

1.  The  first  term  of  an  increasing  progression  is  4,  the  com- 
mon difference  3,  and  the  number  of  terms  10  :  what  is  the 
last  term  ? 

Analysis. — By  considering  the  manner  in  which  the  increasing 
progression  is  formed,  we  see  that  the 
2d   term    is  obtained    by  adding   the  operation. 

common   difference  to  the   1st  term  ;  9  no.  less  1 

the  3d,  by  adding  the  common  differ-  3  com.  diff. 

ence  to  the  2d  ;  the  4th,  by  adding  the  27 

common  difference  to  the   3d,  and  so  4  1st  term, 

on;  the  number  of  additions^  in  every  31  last  term, 

case^  being  1  less  than  the  ymmber  of 

terms  found.  Instead  of  making  the  additions,  we  may  multiply  the 
common  difference  by  the  number  of  additions,  that  is,  by  1  less  than 
tlie  number  of  terms,  and  add  the  first  term  to  the  product. 

Rule. — Multiply  the  common  difference  by  1  less  than  the 
mimber  of  tei-ms  :  if  the  2^^'ogression  is  increasing,  add  the  pro- 
duct to  the  first  term,  and  the  sum  'will  be  the  last  term  ;  if  it  is 
decreasing,  subtract  the  product  from  the  first  term  and  the  dif- 
ference will  be  the  last  term. 

EXAMPLES. 

1.  What  is  the  18th  term  of  an  arithmetical  progression,  of 
which  the  fii'st  term  is  4,  and  the  common  difference  5  ? 

2.  A  man  is  to  receive  a  certain  sum  of  money  in  12  pay- 
ments :  the  first  payment  is  $300,  and  each  succeeding  pay- 
ment is  less  than  the  previous  one  by  $20  :  what  will  be  the 
last  payment  ? 

3.  What  will  $200  amount  to  in  15  years,  at  7  per  cent 
simple  interest :  the  first  year  it  increases  $14,  the  second,  $28, 
and  so  on  ? 

320.  When  you  know  the  first  term,  the  common  difference  and  the 
iiuiuber  of  terms,  how  do  you  find  the  last  term  1 


820  ARITHMETICAL    PROGRESSION. 

4.  A  man  bus  a  piece  of  land  35  rods  in  length,  which 
tapers  to  a  point,  and  is  found  to  increase  ^  rod  in  width,,  for 
every  rod  in  length :  what  is  the  width  of  the  wide  end  ? 

5.  James  and  John  have  100  marbles.  It  is  agreed  between 
them  that  John  shall  have  them  all,  if  he  Avill  place  them  in  a 
straight  line  half  a  foot  apart,  and  so  that  he  shall  be  obliged 
to  travel  300  feet  to  get  and  bring  back  the  farthest  marble ; 
and  also,  if  he  will  tell,  without  measuring,  how  far  he  must 
travel  to  bring  back  the  nearest. 

CASE   II. 

321.  Knoiving  the  two  extremes  of  an  arithmetical  'progreS' 
sion  and  the  number  of  terms,  to  find  the  common  difference. 

1.  The  two  extremes  of  a  progression  are  4  and  68,  and  the 
number  of  terms  17  :  what  is  the  common  difference  ? 

Analysis. — Since  the  common  difference  multiplied  by  1  less  than 
the  number  of  terms  gives  a  product  equal 
to  the  difference   of   the  extremes,   if  we  operation. 

divide  the  difference  of  the  extremes  by  1  68 

less  than  the  number  of  terins^  the  quo-  4 

Xieni  vf\\\  he  WiQ  common  difference :  hence,         17  —  1  =  16)64(4 

Rule. — Subtract  the  less  extreme  from  the  greater,  and  divide 
the  remainder  by  1  less  than  the  number  of  terms  :  the  quotient 
will  be  the  common  difference. 

EXAMPLES. 

1.  A  man  started  from  Chicago  and  travelled  15  days ;  each 
day's  journey  was  increased  by  the  distance  which  he  travelled 
the  first  day  :  what  was  his  daily  increase  if  he  travelled  75 
miles  the  last  day  ? 

2.  A  merchant  sold  14  yards  of  cloth,  in  pieces  of  1  j-ard 
each  ;  for  the  first  yard  ,he  received  $1,  and  for  the  last  $2G^  . 
what  was  the  diflference  in  the  price  per  yard  ? 


321.  ^Vllrn  you  know  the  extremes  and  number  of  terms,  how  do  you 
find  tile  oonniion  dLfTercujio  1 


2  5  8 
.23  20  17 

11 
14 

14 
11 

17 
8 

20 
5 

23 
2 

25  25  25 

25 

25 

15 

25 

25 

ARITHMETICAL    PROGRESSION.  321 

3.  A  board  is  17  feet  long:  it  is  2-^  inches  wide  at  one  end, 
and  144  at  the  other :  what  is  the  average  increase  in  width 
per  foot  in  length  ? 

CASE   III. 

322.  To  find  the  sum  of  the  terms  of  an  arithmetical  prO' 
gression. 

1.  What  is  the  sum  of  the  series  whose  first  term  is  2,  com- 
mon difference  3,  and  number  of  terms  15  ? 
Given  series, 
Same,  order  inverted. 
Sum  of  both  series. 

Analysis. — The  two  series  are  the  same  ;  hence,  their  sum  is 
equal  to  twice  the  given  series.  But  their  sum  is  equal  to  the  sum 
of  the  two  extremes,  2  and  23.  taken  as  many  times  as  there  are 
terms  J  and  the  given  series  is  equal  to  half  this  sum,  or  to  the  sura 
of  the  extremes  multiplied  by  half  the  number  of  terms. 

Rule. — Add  the  extremes  together  and  multi2'>Iy  their  sum  by 
half  the  number  of  terms  ;  the  product  will  he  the  sum  of  all  the 
t4rms. 

EXAMPLES. 

1.  What  debt  could  be  discharged  in  a  year,  by  weekly  pay- 
ments in  arithmetical  progression,  the  first  payment  being  $5, 
and  the  last  $100  ? 

2.  A  person  agreed  to  build  56  rods  of  fence  ;  for  the  first 
rod  he  was  to  receive  6  cents,  for  the  second,  10  cents,  and  so 
on  :  what  did  he  receive  for  the  last  rod,  and  how  much  for  the 
whole  ? 

3.  If  a  person  travel  30  miles  the  first  day,  and  a  quarter  of 
a  mile  less  each  succeeding  day,  how  far  will  he  travel  in  30 
days  ? 

4.  If  120  stones  be  laid  in  a  straight  line,  each  at  a  distance 
of  a  yard  and  a  quarter  from  the  one  next  to  it,  how  far  must  a 
person  travel  who  picks  them  up  singly  and  places  them  iu  a 

a23.  How  do  you  find  the  sum  of  the  series  1 


822  ARITHMETICAL   PEOGEESSION. 

heap,  at  the  distance  of  6  yards  from  the  end  of  the  line  and  in 
its  continuation  ? 

CASE   IV. 

323.  Having  given  the  first  and  last  terms,  and  the  common 
difference,  to  find  the  number  of  terms. 

1.  The  first  term  of  an  arithmetical  progression  is  5,  the  com- 
mon difference  4,  and  the  last  term  41  :  what  is  the  number  of 
terms  ? 

Analysis. — Since  the  last  term  is  equal  operation. 

to  the  first  term  added  to  the  product  of  the         41  —  5  =  36 
common  dilTerence,  by  1  less  than  the  num-  4)36(  =  9 

ber  of  terms  (Art.  320),  it  follows  that,  if  9  +  1  =  10  No.  terms, 
the  first  term  be  taken  from  the  last  term, 

the  difference  will  be  equal  to  the  product  of  the  common  difference 
by  1  less  than  the  number  of  terms  :  if  this  be  divided  by  the  com- 
mon difference,  the  quotient  will  be  1  less  than  the  number  of  terms.' 

Rule. — Divide  the  difference  of  the  tico  extremes  hy  the  com- 
mon difference,  and  add  1  to  the  quotient :  the  sum  will  be  the 
number  of  terms. 

EXABIPLES. 

1.  A  farmer  sold  a  number  of  bushels  of  wheat ;  it  was 
agreed  that  for  the  first  bushel  he  should  receive  50  cents,  and 
an  increase  of  9  cents  for  each  succeeding  bushel,  and  for  the 
last  he  received  $500  :  how  many  bushels  did  he  sell  ? 

2.  A  person  proposes  to  make  a  journey,  and  to  travel  15 
miles  the  first  day,  and  33  miles  the  last,  with  a  dally  increase 
of  11  miles  :  in  how  many  days  did  he  make  the  journey,  and 
what  was  the  Avhole  distance  travelled  ? 

3.  I  owe  a  debt  of  $2325,  and  wish  to  pay  it  in  equal  install- 
ments, the  first  payment  to  be  $575,  the  second  $500,  and 
decreasing  by  a  common  diflcrencc,  until  the  last  payment  which 
is  $200  :  what  will  be  the  number  of  installments  ? 


323.  Having  given  the  first  and  last  terms  and  the  common  difference, 
Low  do  \ou  find  the  number  of  terms  ! 


GEOMETRICAL   PKOaKESSIOJ^.  328 


GEOMETRICAL  PROGRESSION. 

324.  A  Geometrical  Progression  is  a  series  of  terms, 
each  of  which  is  derived  from  the  preceding  one,  by  muUiply- 
ing  it  by  a  constant  number.  The  constant  multiplier  is  called 
the  ratio  of  the  progression. 

32.5.  If  the  ratio  is  greater  than  1,  each  term  is  greater  than 
the  preceding  one,  and  the  series  is  said  to  be  increasing. 

If  the  ratio  is  less  than  1,  each  term  is  less  than  the  preced- 
ing one,  and  the  series  is  said  to  be  decreasing  ;  thus, 

1,     2,     4,     8,  16,  32,  &c. — ratio  2 — increasing  series  : 
32,  16,    8,     4,    2      1,  &c. — ratio  \ — decreasing  series. 

The  several  numbers  resulting  from  the  multiplication  arc 
called  terms  of  the  progression.  The  first  and  last  are  called 
the  extremes,  and  the  intermediate  terms  are  called  means. 

326.  In  every  Geometrical,  as  well  as  in  every  Arithmetical 
Progression,  there  are  five  parts  : 

1st :    The  first  term  ; 

2d  :    The  last  term  ; 

3d  :     The  common  ratio  ; 

4th  :   The  number  of  terms  ; 

5th  :   The  sum  of  all  the  terms. 

If  any  three  of  these  parts  are  known,  or  given,  the  remain- 
ing ones'  can  be  determined. 

324.  What  is  a  geometrical  progression  1  What  is  the  constant  multi- 
plier called  1 

325.  If  the  ratio  is  greater  than  1,  how  do  the  terms  compare  with  each 
other  1  What  is  the  series  then  called  !  If  the  ratio  is  less  than  1,  how 
do  they  compare  T  What  is  the  series  then  called  ?  What  are  the  several 
numbers  called  ?  What  are  the  first  and  last  terms  called  1  What  are  tho 
intermediate  ones  called  l 

326.  How  many  parts  are  there  in  every  geometrical  progression  \ 
What  are  they "!  How  many  must  be  known  before  'lie  others  can  be 
found  \ 


324  GEOMETKICAL   PROGRESSION. 

CASE   I. 

327.  Having  given  the  first  term,  tJie  ratio,  and  the  number 
of  terms,  to  find  the  last  term. 

1.  The  first  term  is  4,  and  the  common  ratio  3 :  what  is  the 
5th  term? 

Analysis. — The  second  term  is  formed  by  operation. 

multiplying  the  first  term  by  the  ratio;  the     3  X  3  X  3  x  3  =  81 
third  term  by  multiplying  the  second  term  4 

by  the  ratio,  and  so  on ;  the  number  of  mul-  Ans.  324 

tiplications  icing   1   less  than  the  number  of 
terms :  thus,    ■ 

4  =       4,  1st  term, 
3x4=     12,  2d  term, 
3x3x4==     36,  3d  term, 
3x3x3x4=   108,  4th  term, 
3x3x3x3x4=  324,  5th  term. 

Therefore,  the  last  term  is  equal  to  the  first  term  multiplied 
ly  the  ratio  raisvd  to  a  jioioer  whose  cxpomnt  is  1  less  than  the 
number  of  terins. 

Rule. — Raise  the  ratio  to  a  poicer  tchose  exponent  is  1  less 
than  the  number  of  terms,  and  then  midtiphj  this  power  by  the 
first  term, 

EXAMPLES. 

1.  The  first  term  of  a  decreasing  progression  is  2187  ;  the 
ratio  is  l,  and  the  number  of  terms  8  :  what  is  the  last  term  ? 

Note, — The  7lh  power  of  -J-  i.s  -rrx^y;  this  operation. 

multiplied   by  the   first  term,  2187,  gives  1,  (-J-)'  =  •jxVt 

the  last  term.  (ttVt  x  2187  =  1. 

2.  The  first  term  of  an  increasing  geometrical  series  is  8, 
the  ratio  5  :  wliat  is  the  9th  term  ? 

3.  The  first  term  of  a  decreasing  geometrical  series  is  729, 
the  ratio  i  :  what  is  the  10th  term  ? 

M'Zl.  Knowing  the  first  term,  the  ratio,  and  the  number  of  tenns,  how  Jo 
you  find  the  last  term  ? 


OKOMETRICAL    PROGRESSION.  325 

4.  If  a  farmer  should  sell  15  bushels  of  wheat,  at  1  mill  for 
the  first  bushel,  1  cent  for  the  second,  1  dime  for  the  third,  and 
60  on  ;  what  would  he  receive  for  the  last  bushel  ? 

5.  A  man  dying  left  5  sons,  and  bequeathed  his  estate  in  the 
following  manner  ;  to  his  executors  $100  ;  his  youngest  son  was 
to  have  twice  as  much  as  the  executors,  and  each  son  to  have 
double  the  amount  of  the  next  younger  brother :  what  was  the 
eldest  son's  portion  ? 

6.  A  merchant  engaging  in  business,  trebled  his  capital  once 
in  4  years  ;  if  he  commenced  with  $2000,  what  will  his  capital 
amount  to  at  the  end  of  the  12th  year  ? 

7.  A  farmer  wishing  to  buy  16  oxen  of  a  drover,  finally 
agreed  to  give  him  for  the  whole  the  cost  of  the  last  ox  only. 
He  was  to  pay  1  cent  for  the  first,  2  cents  for  the  second,  and 
doubling  on  each  one  to  the  last :  how  much  would  they  cost 
him  ? 

CASE   11. 

328.  Knowing  the  two  extremes  and  the  ratio,  to  Jind  the  sum 
of  the  terms. 

1.  What  is  the  sum  of  the  terms  of  the  progression  2,  C,  18, 
54,162? 

OPERATION. 

6  +  18  +  54  +  162  +  486  =  3  times. 

2  +  6  +  18  +  54  +  162 =z  1  time. 

486  -  2  =  2  times. 
486  -  2      484 


2 


=  242  sum. 


Analysis. — If  we  multiply  the  terms  of  the  progression  by  the 
ratio  3.  we  have  a  second  progression,  6,  18,  54,  162,  486,  which  is 
3  times  as  great  as  the  first.  If  from  this  we  subtract  the  first,  the 
remainder.  486  —  2,  will  be  2  times  as  great  as  the  first;  and  if  this 
remainder  be  divided  by  2.  the  quotient  Avill  be  the  sum  of  the  terms 
of  the  first  progression. 

328.  Knowing  the  two  extremes  and  the  ratio,  how  do  you  find  the  cum 

of  the  terms ! 

IS 


g26  GEOMETlilCAL    PKOGKESSION. 

But  48P  is  the  product  of  the  last  term  of  the  given  progression 
multiplied  by  the  ratio,  2  is  the  first  term,  and  the  divisor  2,  1  less 
than  the  ratio  :  hence, 

Rule. — Multiply  the  last  term  by  the  ratio ;  take  the  differ- 
ence between  this  2}roduct  and  the  first  term  and  divide  the  remairir- 
der  by  the  difference  between  1  and  the  ratio. 

Note. — When  the  progression  is  increasing,  the  first  term  is  sub- 
tracted from  the  product  of  the  last  term  by  the  ratio,  and  the  di\'isor 
is  found  by  subtracting  1  from  the  ratio.  When  the  progression  is 
decreasing,  the  product  of  the  last  term  by  the  ratio  is  subtracted 
from  the  first  term,  and  the  ratio  is  subtracted  from  1 . 

EXAMPLES. 

1.  The  first  term  of  a  progression  is  4,  the  ratio  3,  and  the 
last  term  78722  :  what  is  the  sum  of  the  terms  ? 

2.  The  first  term  of  a  progression  is  1024,  the  ratio  i,  and 
the  last  term  4  :  wliat  is  the  sum  of  the  series  ? 

3.  What  debt  can  be  discliarged  in  one  year  by  monthly 
payments,  the  first  being  $2,  the  second  S8,  and  so  on  to  the 
end  of  the  year,  and  what  will  be  the  last  payment  ? 

4.  A  gentleman  being  importuned  to  sell  a  fine  horse,  said 
that  he  would  sell  him  on  the  condition  of  receiving  1  cent  for 
the  first  nail  in  his  shoes,  2  cents  for  the  second,  and  so  on, 
doubling  the  price  of  every  nail :  the  number  of  nails  in  each 
shoe  being  8,  how  much  would  he  receive  for  his  liorse  ? 

5.  A  laborer  agreed  to  thresh  64  days  for  a  farmer  on  the 
condition  that  he  should  give  him  1  grain  of  wheat  for  the  first 
day's  labor,  2  grains  for  the  second,  and  double  each  succeeding 
day :  what  number  of  bushels  would  lie  receive,  supposing  a 
pint  to  contain  7G80  grains,  and  wliat  number  of  ships,  each 
carrying  1000  tons  burden,  might  be  loaded,  allowing  40  bushels 
to  a  ton  ?  ( 


ANALYSIS. 


327 


ANALYSIS. 

J.  If  12  yards  of  cloth  cost  S48,36,  what  will  7  yards  cost? 

Analysis. — One  yard  of  clolh  will  cost  -^  as  much  as  12  yards  : 
•ince  12  yards  cost  S48,36,  one  yard  will  cost  Jg-  of  $48,36  =  $4,03 
7  yards  will  cost  7  times  as  much  as  1  yard,  or  7  times  -^  of  $48, 3G 
=  828.21  ;  therefore,  if  12  yards  of  cloth  cost  -148,36,  7  yards  will 
cost  $28,21. 

OPERATION. 

4,03 
4,03 

i$,$$      1      7     ^„^^,         1^ 


^S^^i 


$28,21  ;  or 


MM 
7 


I  $28,21  Ans. 

3.  If  27  pounds  of  butter  will  buy  45  pounds  of  sugar,  how 
much  butter  will  buy  36  pounds  of  sugar  ? 

Analysis. — One  pound  of  sugar  will  buy  -^  as  much  butter  as  45 
pounds,  or  -^-^  of  27lbs.  of  batter;  3fc  pounds  of  sugar  will  buy  36 
times  as  much  butter  as  1  pound  of  sugar,  or  36  times  -^  of  27/65., 
which  is  ^^Ibs.  =  21|/65. 

OPERATION. 


^:^        1        36 


3.  "What  will  6|  cords  of  wood  cost,  if  2^  cords  cost  $7^  ? 

Analysis. — Since  2|  cords  =  ^  cords  of  wood  costs  S7|-  ~.  S-^ 
one    cord    will    cost  as  many  dollars  as  -l^  is  contained  times  in  'y 
or  $3  :  C-|  cords  =  ?j?  cords,  will  cost  ^-^  times  as  much  as  1  Qmd 
that  is.  $3  X  3J  =  $8ji  ::=  $20.25. 

0PER.4TI0N. 


108 

5 

27 

.     =2ims.;    or 
0 

108 

21^/55- 

a  ■  ^  ii)  ^  4  ~  4 


$201;    or, 


$ 

4 

$ 
27 

4       81,00 

$20,25  Ans. 


12 

3 

$ 

^ 

n 

$ 

5 

7 

180,00 

328  ANALYSIS. 

Note. — The  fractional  pirt  of  a  dollar  may  always  be  reduced  to 
cent.s  by  annexing  two  ciphers,  and  to  mills  by  annexing  three,  and 
then  dividing  by  the  denominator. 

4.  A  farmer  sold  a  number  of  cows,  and  had  12  left,  which 
was  i  of  the  number  sold  ;  if  the  number  sold  be  divided  by 
I  of  91,  the  quotient  will  be  1  the  number  of  dollars  he  received 
per  head  :  how  much  did  he  receive  per  head  for  his  cows  ? 

Analysis. — 12  is  ^  of  3  times  12  =  36,  the  number  of  cows  sold ; 
36  divided  by  |  of  9J=7,  the  quotient,  8^,  is  |  of  5  times  ^=-i-fA 

=  S25f 

OPERATION. 


12       3         4x3       5      180     ^^,^  „ 

T^1^3^28^i=-T==^2^^'^^^; 


$25,71|-  Ans: 

5.  What  will  20  bushels  of  barley  cost,  in  dollars  and  cents, 
at  7  shillings  a  bushel,  New  York  currency  ? 

Notes. — 1.  Although  United  States  money  is  expressed  in  dollars, 
cents,  and  mills,  still  in  most  of  the  States  the  dollar  (always  valued 
at  100  cents)  is  sometimes  reckoned  in  pounds  shillings  and  pence; 
thus. 

2.  In  the  New  England  States,  in  Indiana,  Illinois,  Missouri, 
Virginia,  Kentucky,  Tennesce,  Mississippi,  and  Texas,  the  dollar  is 
reckoned  at  6  shillings ;  in  New  York,  Ohio,  and  Michigan,  at  8  shil- 
lings ;  in  New  Jersey,  Pennsylvania,  Delaware,  and  Maryland,   at 

•   75.  Gd.;  in  South  ^Carolina  and  Georgia,  at  45.  8fZ. ;  in  Canada  and 
Nova  Scotia,  at  0  shillings. 

3.  It  often  occurs  that  the  retail  price  is  given  in  shillings  and 
pence,  and  the  result  or  cost  is  required  in  dollars  and  cents. 

Analysis. — Since  1  bu.shel  of  barley  costs  7  shillings,  20  busliela 
will  cost  20  times  7  shillings,  or  140  shillings;  and  as  8  shillings 
make  1  dollar,  New  York  currency,  there  will  be  as  many  dollars  ub 
}<  i.^  contained  times  140  =  $17  50. 


ANALYSIS. 


829 


OPERATION. 


20  X  7  -i-  8  =  $171 ;     or 


n 


35,00 

"$17,50  Ans. 


6.  What  will  be  the  cost  of  72  bushels  of  potatoes,  at  35.  M. 
per  bushel,  New  York  currency  ? 


4 

13 

Ans. 

OPERATION. 

Or, 

^0^ 

3 

n 

39 

4 

117,00 

4 

117,00 

$29,75 

$29,75  Ans 

Note. — When  the  pence  is  an  aliquot  part  of  a  shilling,  the  price 
may  be  reduced  to  an  improper  fraction,  which  will  be  the  multiplier 
m  the  denomination  of  shillings ;  thus,  35.  3c?.  =  'i\s.  =  ^^^5. :  or,  the 
shillings  and  pence  maybe  reduced  to  pence;  thus,  3s.  3rf.  =  39rf., 
tn  which  case  the  product  will  be  pence,  and  must  be  divided  by  96, 
the  number  of  pence  in  $1. 

7.  What  will  121  pounds  of  tea  cost  at  65.  Sc?.  a  pound, 
Pennsylvania  currency  ? 


OPERATION. 


3  ^ 

20 

9 

100 

$111  Ans. 


Or, 


00 
9 


25 

4 

100,00 


$11,11j    \ns 


Note. — In  the  last  example  the  multiplier  is  65.  8J.  =  6>5.  —  \ 
or  %Qd.     The  divisor  is  75.  Gc/.  =  7^.s.  =  ^s.^  or  90d     Hence,  to  fiii 
fjie  cost  of  articles  in  dollars  and  cents,  when  the  price  is  in  shillings 
and  pence, 

Muliiplij  the  commodity  hy  the  price,  and  divide  the  product 
by  the  value  of  a  dollar,  ex2')resscd  in  the  unit  of  the  price. 


330 


ANALYSIS. 


8.  How  many  daj^s  work  at  10s.  6c/.  a  day,  must  be  given 
for  18  bushels  of  corn  at  os.  lOd.  a  bushel? 


0 


1$ 

2 


OPERATION. 


Or, 


m 


10  days  Ans. 


10 


10  days  Ans. 

Note. — The  same  rule  applies  in  this  as  in  the  preceding  examples, 
except  that  the  divisor  is  the  price  of  the  articles  received  in  pay- 
ment, reduced  to  the  same  unit  as  the  price  of  the  article  bought. 

9.  What  will  5cwt.  of  coffee  cost,  at  I5.  4c?.  per  pound,  New 
York  currency? 

OPERATION, 

> 

25 

4 


t 


Or, 


n 


250 


25 


250,000 


883,3331  Ans. 


$83,3331  Ans. 

Note. — Reduce  the  cwts.  to  Ihs.  by  multiplying  by  4,  and  then  by 
25,  after  which  proceed  as  in  the  preceding  examples. 

10.  A  merchant  bought  a  number  of  bales  of  cloth,  each 
containg  1331  yards,  at  the  rate  of  12  yards  for  $11,  and  sold 
it  at  the  rate  of  8  yards  for  $7,  by  which  he  lost  $100  in  the 
trade  :  how  many  bales  were  there  ? 

Analysis. — Since  he  paid  Su  for  12  yards,  for  1  yard  he  paid  -A 
of  $11,  or  W  of  %\  \  and  since  he  received  $7  for  8  yard.«.  for  1  yard 
he  received  \  of  $7,  or  \  of  $1,  He  lost  on  1  yard  the  difference 
between  \\  and  |-  —  -jjtj.  of  a  dollar.  Since  his  whole  loss  was  Si 00, 
he  had  as  many  yards  as  ^  is  contained  times  in  100  =  2400  yards; 
and  1  here  were  as  many  bales  as  133J  (the  number  of  yards  in  ] 
bale)  is  contained  times  in  the  whole  number  of  yards  =  18  bales. 


OPERATION. 


_1_ 

'2  4 


(100  -  ^v)  -  133'^  =  18  ^«5.  ^'''    m_ 


100 


6 


3 

18  bales  Ano. 


ANALYSIS.  831 

11.  A  can  mow  an  acre  of  grass  in  7^  hours  ;  B,  in  5  hours  ; 
C,  in  5|  hours  ;  how  many  days  working  6|-  hours,  would 
they  requix'e  to  mow  lo|  acres  ? 

Analysis. — Since  A  can  mow  an  acre  of  grass  in  7^  hours, 
B,  in  5  hours,  and  C  in  5|,  A  can  mow  ^,  B,  \,  and  C.  -^-^  of 
an  acre,  in  1  iiour.  Together,  they  can  mow  t'5+i+4^^=-4l  "'  "•' 
acre  in  1  hour:  and  to  mow  1  a^.TC,  they  will  require  as  many  Imurs- 
as  |-|  is  contained  times  in  1  = -ff  hours:  to  mow  13-|-  acres,  they 
will  require  13-|  times  -1^  =  27  hours,  and  working  6|-  hours  each 
day.  will  require  4  days. 

OPKRATION. 


15       5        45       4o  fit 

Or         ^ 

00     4      , ,         ^''    n 


X  -—  X 


n     $ 


—  =    4  days 


4 


4  days  Ans. 

12.  A  person  employed  three  men,  A,  B,  and  C,  to  do  a 
piece  of  work  for  $132,66.  A  can  do  the  work  alone  in  23^ 
days,  working  12  hours  a  day;  B  can  do  it  in  25  days,  working 
8  hours  a  day;  and  C  can  do  it  in  16  days,  working  IIJ 
hours  a  day.  In  what  time  can  the  three  do  it,  working  to- 
gether, 10  hours  a  day,  and  what  share  of  the  money  should 
each  receive  ? 

Analysis. — Since  A  can  do  the  work  in  23^  days,  working  12 
hours  each  day:  B,  in  25  days,  working  8  hours  each  day;  and  C, 
in  16  days,  working  lli  hours  each  day,  A  can  do  the  same  v/ork  in 
280  days.  B,  in  200  days,  and  C.  in  180  days,  working  1  hour  each 

day  :  then  A,  B,  and  C,  can  do  -^h  +  nh  +  li o  =  Trifo  ^^  ^^^^ 
work  in  1  day,  working  1  hour  :  by  working  10  hours,  they  will  do 
10  times  as  much  ;  or,  the  work  done  by  each  in  1  day  of  10  hours, 
will  be  denoted  by  -^q.  -^^q,  and  ,^^  :  and  the  whole  work  done  in 
1  day  by  tt^tt^  ;  hence,  the  number  of  days  will  be  denoted  bv  the 
number  of  times  which  1  contains  ■jWiT^'We'^  "'''"g^  days. 

If  the  part  which  each  does  in  1  day  be  multiplied  by  the  number 
of  days,  viz.,  7^,  the  product  will  be  the  part  done  by  each  ;  viz.,  Aj 

5^0  x'^W^iV^;  B>  ^x  '  sV^tVs  ■■  '^"d  C,  J^o_.x7^=^%  ;  there- 
fore, A  must  have  ^-^,  B,  ^-^  and  C   J^  of  S132,GG. 


332  ANALYSIS. 

OPERATION. 

First:  l-^Y^=.W7f=V^days.  Ans. 

Second:  8132,66 XTV8  =  ^33,53f|=A's  share. 
$132,66  xf-^=S46,95fi=B's    share. 
8132,76  X  ^^=$52,16|^=:C's   share. 
Total  paid     3132  66" 

13.  If  336  men,  in  5  days,  working  10  hours  each  day,  can  dig 
a  trench  of  5  degrees  of  hardness,  70  yards  long,  3  yards  wide,  and 
2  yards  deep ;  how  many  days  of  12  hours  each,  will  240  men 
require  to  dig  a  trench  36  yards  long,  5  yards  wide,  and  3  yards 
deep,  of  6  degrees  of  hardness  ? 

Analysis. — Since  336  men  require  5  days  of  10  hours  each,  to  dig 
a  trench,  it  will  take  1  man  336  times  5  days  of  10  hours  each,  and 
10  times  (336  X  5)  days  of  1  hour  each^  to  dig  the  same  trench.  To 
dig  a  trench  1  yard  loug.  will  require  -^  as  much  time  as  to  dig  one 
70  yards  long ;  to  dig  one  1  yard  wide,  ^  as  much  as  3  yards  wide; 
to  dig  one  1  yard  deep,  -^  as  much  as  2  yards  deep  ;  and  to  dig  one 
of  1  degree  of  hardness  \  as  much  as  to  dig  one  of  5  degrees  of 
hardness.  240  men  can  dig  a  trench  1  yard  long,  1  yard  wide.  1  yard 
deep,  and  of  1  degree  of  hardness  in  ^^  of  the  time  that  1  man  can 
dig  the  same,  and  in  -jlj  as  many  days  of  12  hours  each,  as  of  1  hour 
each;  but  to  dig  one  36  yards  long,  will  require  36  times  as  much 
time  as  to  dig  one  1  yard  long ;  to  dig  one  5  yards  wide,  5  times  as 
much  as  1  yard  wide ;  to  dig  one  3  yards  deep,  3  times  as  much  as 
1  yard  deep;  and  to  dig  one  of  6  degrees  of  hardness  will  require 
6  times  as  much  time  as  to  dig  one  of  1  degree  of  hardness. 

OPERATION. 

«Ji^5xi0^1xixlxlx-Lxlx?^x*x?x?=9*,. 
1  Tt0    1i    fi    $    t0    1^2     1     1    1    1 

Or, 


U0 

m 

X 

$ 

n 

10   q 

n 

% 

$ 

% 

H 

$ 

0^ 

0  da; 

ANALYSIS. 


083 


Note. — The  principle  of  the  above  analysis  is  this  :  1st.  Find  how 
many  hours  it  will  take  1  man  to  dig  1  cubic  yard  of  trench  ;  this  ia 
done  in  the  first  part  of  the  analysis.  2d.  Find  how  long  it  will  take 
240  men.  working  12  hours  a  day,  to  dig  the  required  trench,  working 
at  the  same  rate ;  this  is  done  in  the  second  part  of  the  analysis. 

14.  If  20  cords  of  wood  are  equal  in  value  to  6  tons  of  hay, 
and  5  tons  of  luiy  to  36  bushels  of  wheat,  and  12  bushels  of 
wheat  to  25  bushels  of  com,  and  14  bushels  of  corn  to  5Q 
pounds  of  butter,  and  72  pounds  of  butter  to  8  days  of  labor ; 
how  many  cords  of  wood  will  be  equal  to  IG  days  of  labor? 

Analysis. — Since  20  cords  of  wood  arc  equal  to  6  tons  of  hay, 
1  Ion  of  hay  is  equal  to  ^  of  20  cords  of  wood,  or  ^-  cords  ;  5  tons 
are  equal  to  5  times  ^^  or  ^^  cords  ;  since  5  tons  of  hay.  or  ^^  of  a 
cord  of  AA"ood  are  equal  to  36  bushel.s  of  wheat,  1  bushel  of  wheat  is 
equal  to  ^-^  of  ^^  =  -|f  cords,  and  12  bushels  of  wheat  are  equal  to 
12  limes  |^  =  ^J  cords  ;  and  since  12  bushels  of  wheat,  or  ^^  cords 
of  wood  are  equal  to  25  bushels  of  corn,  1  bushel  of  corn  is  equal  to 
^V  ^^  y  =  t  of  a  cord  of  wood,  and  14  bushels  of  corn  are  equal  to 
14  times  f  ~  2_8  cords  ;  and  since  14  bushels  of  corn,  or  2_8  cords  of 
wood  are  equal  to  56  pounds  of  butter,  1  pound  of  butter  is  equal 
Jg.  of  2J  =  ^ig^  of  a  cord,  and  72  pounds  of  butter  are  equal  to  72 
times  ^Ig  =  4  cords  of  wood  ;  and  since  72  pounds  of  butter  or  4  cords 
of  wood  are  equal  to  8  days'  labor,  1  day's  labor  is  equal  to  -I-  of 
A  =:z  ^  cord  of  wood,  and  16  daj's  labor  are  equal  to  16  times  -^  of  a 


cord,  or  8  cords  of  wood. 


20 
1 


1 
6 


5 
1 


1 
36 


.|    Xf>XiXor?Xi    X4)--Xt    X 


12 

T 


1^ 

25 


OPERATION. 

14     2 
1  ^"56"^  1 


X^X8XY=8cord3;or, 


0 

i0  ^ 

$0 

$ 

t$ 

n 

$$ 

u 

0 

n^ 

10 

8  cords.  Ans. 

Note. — This  and  similar  examples  fall  under  what  is  called  the 
Chain  Rule.  In  analyzing,  then,  always  commence  with  the  ternj 
which  is  of  the  same  name  ur  kind  as  the  required  answer. 


334  ANALYSIS. 

15.  A,  B,  and  C,  put  in  trade  $5626 :  A's  stock  was  in  5 
months,  B's,  7  months,  and  C's,  9  months.  They  gained  $1260, 
wliich  was  so  divided  that  A  i-eceived  $4  as  often  as  B  had  $5; 
and  as  often  as  C  had  $3.  After  receiving  82164,50,  B  ab- 
sconded. What  was  each  one's  stock  in  trade,  and  how  much 
did  A  and  C  gain  or  lose  by  B's  withdrawal  ? 

Analysis. — Since  A  received  S4  as  often  as  B  had  So,  and  as  often 
aS  C  had  $3,  if  the  whole  gain  were  divided  into  12  equal  parls,  A 
would  have  ^.  B,  ^j.  and  C.  y%,  of  $1260,  or  A  would  have  S420, 
B,  $525.  and  C.  $315.  Now.  if  their  respective  gains  be  divided  by 
the  luimber  of  months  each  one's  stock  continued  in  trade,  the  quo- 
tients will  represent  their  monthly  gains,  viz.,  A's  will  be  $420 -r  5 
=  S84  ;  B's,  $525  -r  7  =  $75  ;  and  C's.  $315  -r  9  =  $35,  which 
gives  $194  as  their  whole  gain  for  1  month. 

But  since  each  one's  share  of  the  gain  for  a  given  time  will  be  to 
the  whole  gain  for  the  same  time,  as  each  one's  stock  to  the  whole 
stock:  it  follows  that,  A  will  have  ^^r-.  B,  J^.  nnd  C,  y^^.  of  the 
whole  stock,  or  A  will  have  $2436,  B,  $2175,  and  C,  $1015.  When 
B  ran  away  he  was  entitled  to  his  original  stock  $21 75.  and  his  share 
of  the  gain  for  7  months,  that  is,  to  $2175  +  $525  =  $2700;  but  as 
he  took  away  only  $2164,50,  A  and  C  gained  $535.50  by  his  with- 
drawal, which  must  be  divided  between  Hicm  in  tlic  ratio  of  their 
'  If 

investments,  or  as  4  to  3  ;  therefore.  A  will  have  4-.  and  C  -f  of  B's 
unclaimed  portion,  or  A  will  have  $306,  and  C  $229.50. 

OPERATION. 
4  +   5   +   3   =   12. 

A's  whole  gain  =  j\  of  SI  2 60  =  $420 
B's  "  "  =  y\  «  «  _  $525 
C's       "        "    =  j%  "        "     =  $315 

A's  monthly  gain  =  $420  -j-  5  =:  $84 
B's  "  "  =  $525  -f  7  =  $75 
C's         "        "      =  $315  —  9  =  $35 

$194 

A's  stock  =  /^  of  $5626  =  $2436 
B's  "  =  yVi  "  "  =^2175 
C's       **     =  T-Vr    "         "     =  $1015 


ANALYSIS.  So5 

$2175  +  $525  -  $2164,50  =  $535,50,  what  B  left, 
i-  of  $535,50  =  $306         A's  share  of  it. 
3.  "         «        =  $229,50    B's  share  of  it. 
16.  Mr.  Johnson  bought  goods  to  the  amount  of  $2400,  1  to 
be   paid  in  3  months,  i  in  4  months,  |  in  6  montlis,  and  the 
remainder  in  8  months  :  what  is  the  equated  time  for  the  pay- 
ment of  the  whole  ? 

Analysis.— SSOO  to  be  paid  in  3  months,  is  the  same  as  $1^  to  bo 
paid  in  2400  months  ;  SGOO,  in  4  months,  the  same  as  Si  in  2400 
months;  $600,  in  6  mouths;  the  same  as  $1,  in  3600  months;  and 
$400  in  8  mouths,  the  same  as  $1,  in  3200  months.  Then  $1,  payable 
in  2400  +  2400  +  3600  +  3200  =  11600  months,  is  the  same  as 
$2400  in  2^0^  of  11600  months,  wliich  is  4|  months  —  4  months  26 
days,  the  equated  time  of  payment. 

OPERATION. 

800  X  3  =  2400 
GOO  X  4  =  2400 
600.x  6  =  3600 
400  X  8  =  3200 


2400  11600 

11600  -^  2400  =  4|wo.  =  ioio.  25da.  Ans. 

17.  What  will  be  the  interest  on  $60,48  for  1  year  3  months, 
at  7  per  cent  ? 

Analysis. — Since  the  interest  on  $1  for  1  year  is  7  cents,  or  seven 
hundredths  of  SI.  the  interest  on  $60,48  for  1  year,  will  be  $60,48 
X  .07  =  $4.2336.  The  interest  for  1  month  wnll  be  ■^\  as  much  as 
for  1  year  or  ^  of  $4.2236  =  ^0.3528,  and  for  \yr.  3mo.  =  15 
months,  it  will  be  15  times  as  much  as  for  1  month,  or  $0,3528  x  15 
=  $5,292. 


OPERATION. 


($60,48  X. 07 -f  12)  X  15  =  $5,292  yl?is.  Or,     i^ 


5,04 

()0,4^ 
•  7 
15 


I  $5,292  Ans. 
18.  What  will  be  the  interest  on  $88,92,  for  8mo.  20Ja.,  at 
7  per  cent? 


836 


ANALYSia. 


Analysis. — Since  the  interest  on  ^1  is  7  cents  for  1  year,  the  in- 
terest  on  $^88.92  for  1  year  will  be  $88.92  x  .07  =  S6.224  ;  the  in- 
terest for  1  month  will  be  ^-^  of  $6,224  =  $0,5187 ;  and  since  the 
number  of  days  divided  by  30  will  give  the  value  of  those  days  ia 
decimals  of  a  month  (Art.  222)  20c/a.  =  .6f  months.  The  interesi 
for  Pmo.  20cZa.  =  8.6|  months,  will  be  8.6|-  times  as  much  as  for  1 
mouth  =  0.5187  X  8.6|  =  $4.4954. 

OPERATION. 

($88,92  X  .07-M2)x8.6f=$4,4954  A?is. 


n 


2  47 

.07  Or,       .It 

26  ^^  $0 


'^"^    26 


I  $4.4954  I  S4.4954  An>s. 

19.  A  liquor  mercliant  mixed  together  25  gallons  of  brandy 
at  $],60  a  gallon,  25  gallons  at  $1,80,  10  gallons  at  $2,50,  and 
20  gallons  of  water ;  wliat  was  the  value  of  1  gallon  of  the 
mixture,  and  what  was  the  gain  on  a  gallon  if  he  sold  it  at  the 
average  price  of  the  liquor  ? 

Analysis. — The  value  of  20  gallons  of  water  would  be  0  ;  of  25 
gallons  of  brandy  at  $1,60  a  gallon  would  be  $1,60  x  25  =  $40;  of 
25  gallons  at  $1.80,  would  be  $1.80  X  2a  =  $45  ;  of  10  gallons  at 
$2,50,  would  be  $2.50  X  10  =  $25.  25  +  25  +  10  gallons  of 
brandy  +  18  gallons  of  water  =  80  gallons,  the  amount  of  the 
mixture;  and  $40  +  $45  +  25  =  $110,  the  value  of  the  mixture; 
hence,  if  80  gallons  are  wortli  $110,  one  gallon  is  worth  -^  of 
$110  =  $1,371.  But  25  +  25+10  =  60  gallons  of  brandy,  arc 
worth  $110,  and  $110  --  60  =  $1.83^,  the  average  price  per  gallon 
of  the  brandy;  therefore  $1,83-J- —  $1,37^  =  45|- cents,  the  gain  oa 
1  gallon. 

OPERATION. 

20  X  0  r^  00 
25  X  l.GO  =  40 
25  X  1.80  -^  45 
10  X  2.50  =  25 

80  no 

8110-^80  =  $l,37.V  value:  ^1 10 -f- G0=r$1.83i  averagre  price, 
81,83^  -  $1,371  =  e0,45;;. 


ANALYSIS. 


337 


10.  A  merchant  has  three  kinds  of  cloth,  worth  $lf,  $2 J, 
fi^^a.  yard :  what  is  the  least  number  of  whole  yards  he  can 
sell,  to  receive  an  average  price  of  $2},  a  yard  ? 

Analysis. — If  he  sells  1  yard  worth  Sl|-,  for  $2 J,  he  will  gain 
j  of  a  dollar;  to  gain  1  dollar  he  must  sell  as  many  yards  as  |  is 
contained  times  in  1,  or  -|  yards.  But  since  he  is  neither  to  gain  or 
lose  by  the  operation,  if  he  gains  on  one  kind,  he  must  lose  an  equal 
'mm  on  some  other ;  hence,  he  must  sell  some  that  is  worth  more 
ihan  the  average  price.  If  he  sell  1  yard  worth  $3|-  for  $2^^,  he  will 
lose  -I  of  a  dollar;  and  to  lose  $1,  he  must  sell  -|-  of  a  yard.  There- 
fore, to  make  the  loss  equal  to  the  gain,  he  must  sell  ^  of  a  yard  at 
$3|-  a  yard,  as  often  as  he  sells  I-  of  a  yard  at  Slf  a  yard. 

If  he  sells  1  yard  Avorth  $2^,  for  $2i,  ho  gains  -^  of  a  dollar,  and 
to  gain  Si  he  must  sell  4  yards ;  hence,  to  keep  the  average  price,  he 
must  lose  as  much  on  some  other,  and  as  he  can  only  lose  on  that  at 
$3-|  a  yard,  he  must  sell  enough  of  that  to  lose  $1,  which  would  be 
|-  of  a  yard  ;  therefore,  as  often  as  he  sells  -|  yards  at  Slf  a  yard,  he 
must  sell  I  yards  at  $3-|  a  yard;  and  as  often  as  he  sells  4  yards  at 
$2^  a  yard,  he  must  sell  -|  yards  at  $3|-  a  yard. 

But  since  it  is  desirable  to  have  the  proportional  parts  expressed  in 
the  least  whole  numbers,  we  may  multiply  the  numbers  by  the  least 
common  multiple  of  their  denominators,  and  divide  the  products  by 
their  greatest  common  factor;  this  being  done,  we  obtain  in  the 
above  example,  3  yards  at  Sl|  a  yard.  10  yards  at  $2^  a  yard,  and 
4  yards  at  $3|  a  yard. 

OPERATION. 

3 

10 

4 

21.  The  hour  and  minute  hands  of  a  clock  are  together  at 
12  o'clock  :  when  are  they  next  together  ? 

Analysis. — Since  the  minute  hand  passes  over  60  minute  spaces 
while  the  hour-hand  passes  over  5.  the  minute-hand  passes  over 
12  minute  spaces  while  the  hour-hand  passes  over  1,  gaining  11 
minute  spaces  on  the  hour-hand  in  12  minutes  of  time  ;  the  minme- 
hand  requiring  one  minute  of  time  to  pass  over  1  minute  of  space. 
Hence,  in  1  minute  of  time,  the  minute-hand  gains  on  the  hour  hand. 
\-^  of  a  minute  space. 


n  ' 

.5 

6 

6 

^ 

4 

20 

20 

4 
5 

4 
5 

4 

4 

8 

OoS  ANALYSIS. 

When  the  minntc-hand  returns  to  12,  the  hour-hand  will  be  at  1, 
and  M"ill  require  the  niinute-hand  to  gain  5  minute  spaces.  As  the 
minute-hand  passes  over  -^^  the  space  gained,  to  gain  5  minute  spaces 
it  must  pass  over  -i^  of  5  =  ^^  =  5^j  minute  spaces,  requirin"!  5^ 
minutes  of  time  =  5ini.  27-^jSCC.^  which  added  to  1  o'clock,  gives 
ihr.  5ini.  27-^^.sec. 

Second  Analysis. — In  12  hours  the  minute-hand  passes  the  hour- 
hand  11  times,  consequently,  if  both  are  at  12,  the  minute-hand  will 
pass  the  hour-hand  the  first  time  in  ^  of  12  hours,  or  ihr.  5mi. 
27^jscc.    It  will  pass  it  the  second  time  in  -^  of  12  hours,  and  so  on. 

OPERATION. 

5  X  -^-f  =  yy  =  h-^m'i.  —.  omi.  27-^scc.,  which  added  to 
Ihr.  =r  l//r.  omi.  27^sec.  Ans. 

21.  An  apple  boy  bouglit  a  certain  number  of  apples  at  the 
rate  of  3  for  1  cent,  and  as  many  more  at  4  for  1  cent,  and 
selling  them  again  at  2  for  1  cent,  he  found  that  he  had  gained 
15  cents  :  how  many  apples  had  he  ? 

Analysis. — Since  he  bought  a  number  of  apples  at  3  for  a  cent, 
and  as  many  more  at  4  for  a  cent,  he  paid  ^  of  a  cent  apiece  for  the 
first,  and  ^  of  a  cent  apiece  for  the  second  Jot :  then,  ^  +  \  =  ^  o( 
a  cent,  what  he  paid  for  one  of  each,  and  -^-t-  2  =  -^  of  a  ceni,  the 
average  price  for  all  he  bought.  Since  he  sold  at  2  for  a  cent,  or  -J-  a 
cent  a  piece,  he  must  have  gained  on  each  apple  the  difference  be- 
tween -^  and  -^j  =  ^  of  a  cent ;  hence,  to  gain  1  cent  he  must  sell 
as  many  apples  as  -^  is  contained  times  in  1  =  4-|  apples,  and  to 
gain  15  cents  he  must  sell  15  times  as  many,  or  4-|  x  15  ^  72  apples. 

operation. 

Ill    7  7_L.9   7  1  7     5 

3^4—  T2'  T^     ■     ^  ■ —  TT'  2  2T  —  2T' 

1  -f  ^\-  =  4|,         4|-  X  15  =  72  apples.  .In*'. 

22.  A  gentleman   left   to  his   three  sons,  whose  ages  were 
13,  15  and   17  years,  $15000,  to  be  divided  in  such  a  maimer, 
that  each  share  being  put  at  interest,   at  7   per  cent,  should 
give  to  each  son  the  same  amount  when  he  attained  the  age  of 
21  years  :  wliat  Avas  the  share  of  each  ? 


ANALYSIS.  839 

ANJi LYSIS. — By  the  question  their  respective  shares  would  be  at 
interest  8,  6  and  4  years. 

Find  the  present  worth  of  $1  for  8,  6  and  4  years,  respectively: 
They  are  S0.G41025G  +,  $0,7042253  +  ,  and  $0,78125.  These  s'nns 
teing  put  at  interest  at  7  per  cent,  will  each  amount  to  $1  at  tho 
expiration  of  their  respective  times  ;  and  the  sum  of  these  numbers, 
$0,6410256  +  $0,7042253+^0,78125  =  $2,1265009+  is  the  amount, 
which  being  so  distributed  among  them,  will  produce  $1  to  each.  If 
each  number  be  divided  by  the  sum,  $2,1265009,  the  quotients  will 
denote  the  parts  of  $1,  which  according  to  the  conditions  of  the 
question,  each  person  should  receive,  and  which  put  at  interest  will 
produce  equal  amounts  at  the  end  of  their  respective  times ;  there- 
fore, each  person  will  receive  for  his  entire  share  15000  like  parts  of 
one  dollar. 

OPERATION. 

$1  -^  1.56  =  $0,6410256  +  present  worth  of  $1  for  8  years. 

$1  -^  1.42  =  $0,7042253  +  "            "             "         6     " 

$1  -f-  1.28  =  $0.78125  "            «             "         4     « 
$2,1265009 

S0,6410256  -f  2.1265  x  15000  -  $4521,694 
$0,7042253  +  2.1265  x  15000  =  $4967,494 
$0,78125      -^  2.1265  x  15000  =  $5510,815 

23.  A,  B,  C,  and  D,  agree  to  do  a  piece  of  work  for  $312. 
A,  B,  and  C,  can  do  it  in   10  days  ;  B,  C,  and  D,  in  7-i-  days  ; 

C,  D,  and  A,  in  8  days  ;  and  D,  A,  and  B,  in  8y  days.  In  how 
many  days  can  all  do  it,  working  together  ;  in  how  many  days 
can  each  do  it  working  alone  ;  and  what  part  of  the  pay  ought 
each  to  receive  ? 

Analysis. — Since  A,  B,  C,  can  do  the  work  in  10  days,  they  can 
do  -^-g  =  -^^  of  it  in  1  day  :  since  B,  C,  D.  can  do  it  in  7-^  days,  they 
can  do  ^^  =  -^^  of  it  in  1  day;  since  C,  D,  A,  can  do  it  in  8  day.s, 
they  can  do  ^  =  ^^^  of  it  hi  1  day;  and  since  D,  A,  B.  can  do  it  in 
8^  days,  they  can  do  -^q  =  -^^  of  it  in   1  day ;  hence.  A,  B,  C,  and 

D,  by  working  3  days  each,  will  do  ^2_  +  ^^  +  ^i^^  +  ^i^  =  -^^^ 
of  the  work,  and  in  1  day  they  will  do  }  of  ^^^  =  J^.  It  will  then 
tiike  them  as  many  days  to  do  the  whole  as  -^-^^  is  contained  times  in 
1   -  6^  days. 


310  ANALYSIS    AND 

By  subtracting,  in  succession,  what  the  three  can  do  in  1  day,  when 
tliey  work  together,  from  what  the  four  can  do  in  1  day,  we  shall 
have  what  each  one  will  do  in  1  day ;  viz.,  J^  —  -^^  =  -jl^,  what 
D  will  do  in  1  day;  ^^  —  I'^o  —  tIo'j  what  A  can  do  in  1  day; 
^ia-tio^  ihy  ^^"li^t  B  can  do  in  1  day ;  J=^  -  t^o  =  Th:  ^'^^^^ 
C  can  do  in  1  day.  It  will  take  each  as  many  days  to  do  the  whole 
work  as  the  part  which  he  can  do  in  1  day  is  contained  times  in  1  ; 
viz.,  1  -I-  y4o  =  40  days.  A's  time  to  do  it;  1  -|-  j^  =  30  days,  B's; 
1  --  ^  =  24  days,  C's  ;   1  -;-  j^  =  17-^  days.  D"s. 

Now.  each  should  receive  such  a  part  of  the  whole  amount  paid, 
viz.,  $312,  as  he  did  of  the  whole  work.  This  part  will  be  denoted 
by  what  he  did  in  1  day  multiplied  by  the  number  of  days  he 
worked:  viz.,    A.  ^X6^%=^-^:  B,  j^^x6^^=^-^;   0,^1^x0^  = 

19-     '''•1^0'^"l9        19- 

OPERATION. 

■j^  =  i\fo5  ^'l^at  A,  B,  C,  does  in  1  day. 
_2_  _  .UL       "BCD      "         " 

8      —    120'  ^'    -^J  -^J 

_7_  _    1 4_        "      71    A    R       "  « 

fiO    —    120'  -^>  -^5   ^> 

t¥o  +  -i^'o  +  T^  +  i'^  =  t¥u'  ^v^at  A,  B,  C,  and  D,  can 
do  in  3  days. 

yYo  -7-  3  r=  jL^y,  what  A,  B,  C,  and  D,  can  do  in  1  day. 
y'^^o  —  y"^  =  jfo^  what  A  can  do  in  1  day  ;  1  -i-  yf ^  —  AOda. 

T2  0  120   —    120  -^J  ■*■     •    TTo   —  ^^""* 

JL9-  _  JIJ_  _  _5_     «      n         u  li  1  _i_    ,5_  _  94  7 

T20  l-'O    —    120  ^»  ■*■      •      120    —   •<i'±((«. 

t2  0  120    —    120  -^J  '120    —  ^'  7     • 

Hence,  the  share  of  each  will  be, 

S312xfiy=§  49,2Gt?^,  A's  share. 
$312Xy't=^$  65,68-1^,  B's  share. 
$3l2xtV  =  ^  S2,l0f||-,  C's  share. 
$!312xTV  =  ^114'94iA    D's  share. 

$312,00  amount  paid  to  A,  B,  C,  and  D. 


PJIOMTSCUOUS    EXAMPLES.  84:1 

24.  A  person  owning  ^  of  a  vessel,  sells  |-  of  his  share  for 
$1736  :  Avhat  was  the  value  of  the  whole  vessel  ? 

25.  If  a  man  performs  a  journey  in  7i  days,  travelling  14^  hours 
a  day,  in  how  many  days  will  he  perform  the  same  journey, 
by  travelling.  10  5^  hours  a  day  ? 

26.  If  ^j-  of  a  pole  stands  in  the  mud,  2  feet  in  the  water,  and 
|-  above  the  water,  what  is  the  length  of  the  pole  ? 

27.  After  spending  i  of  my  money,  and  i  of  the  remainder, 
I  had  $1062  left :  how  much  had  I  at  first  ? 

28.  Suppose  a  cistern  has  two  pipes,  and  that  one  can  fill  it  in 
7J-  hours,  and  the  other  in  4i  hours  :  iu  what  time  can  both  fill 
it  runninjT  together  ? 

29.  If  54  yards  of  ribbon  cost  $9,  what  will  26  yards  cost? 
SO.  If  2  acres  of  land  cost  i  of  f  of  |   of  $300,  what  will 

i  of  -I  of  10|-  acres  cost,'* 

31.  There  is  a  regiment  of  soldiers  to  be  clothed:  each  suit  is 
to  contain  3^  yards  of  cloth  1-|  yards  wide  :  how  much  shal- 
loon that  is  |-  yards  wide  is  necessary  for  lining  ? 

32.  How  much  tea  at  7s.  61:/.  a  pound  must  be  given  for  234 
bushels  of  oats,  at  3s.  9d.  a  bushel.  New  York  currency  ? 

S3.  What  will  3  pipes  of  wine  cost  at  2s.  dd.  per  quart,  New 
England  currency  ? 

34.  A  gives  B  165  yards  of  cotton  cloth,  at  2s.  Qd.  per  yard, 
Missouri  currency,  for  625  pounds  of  lump  sugar :  Irow  much 
was  the  sugar  worth  a  pound  ? 

35.  If  the  expense  of  keeping  1  horse  1  day  is  3s.  AJ.  Canada 
currency,  what  will  be  the  expense  of  keeping  4  horses  3  weeka 
at  the  same  rate  ? 

36.  Bought  10  bales  of  cloth,  each  bale  containing  14  pieces, 
and  each  piece  22^  yards,  at  10s.  8(/.  per  yard,  lUinois  cur- 
rency :  what  was  the  cost  of  the  cloth  ? 

37.  A  has  l\cwt.  of  sugar,  worth  12  cents  a  pound,  for  which 
B  gave  him  12^cut.  of  flour :  what  was  the  flour  worth  a  pound  ? 

38.  What  is  the  value  of  2hhd.  of  molasses,  at  Is.  2d.  per 
quart,  Georgia  currency? 

39.  What  will  be  the  value  of  3  pieces  of  cloth,  each  piece 


342  ANALYSIS   AND 

containing  24^  yards,  at  4s.  Gd.  per  yard,  Pennsylvania  cur- 
rency ? 

40.  Bought  120  yards  of  cloth,  at  65.  8d.  a  yard,  New  York 
currency,  and  gave  in  payment  76  bushels  of  rye,  at  is.  Qd.  a 
bushel,  New  England  currency,  and  the  balance  in  money  :  how 
many  dollars  will  pay  the  balance  ? 

41.  A  merchant  bought  21  pieces  of  cloth,  each  piece  con- 
taining 41  yards,  for  which  he  paid  $1260  ;  he  sold  the  cloth  at 
$1,75  per  yard :  did  he  gain  or  lose,  and  how  much  ? 

42.  The  hour  and  minute  hands  of  a  watch  are  together  at 
12  :  at  what  moment  will  they  be  together  between  5  and  6  ? 

43.  How  many  yards  of  carpeting  f  of  a  yard  wide  will  cover 
the  floor  of  a  room  18  feet  long  and  15  feet  wide  ? 

44.  If  9  men  can  build  a  house  in  5  months,  by  working  12 
hours  a  day,  how  many  hours  a  day  must  the  same  men  work 
to  do  it  in  6  months  ? 

45.  B  and  C  can  do  a  piece  of  work  in  12  days  :  with  the 
assistance  of  A  they  can  do  it  in  9  days :  in  what  time  can  A 
do  it  alone  ? 

46.  A  can  mow  a  certain  field  of  grass  in  3  days,  B  can  do 
it  in  4  days,  and  C  can  do  it  in  5  days  :  in  what  time  can  they 
do  it,  working  together  ? 

47.  Divide  ihe  number  480  into  4  such  parts  that  they  shall 
be  to  each  other  as  the  numbers  3,  5,  7  and  9  ? 

48.  What  length  of  a  board  that  is  8^-  inches  broad,  will  make 
a  square  foot  ? 

49.  The  provisions  in  a  garrison  were  sufficient  for  1800 
men,  for  12  montlis  ;  but  at  the  end  of  3  months,  it  was  rein- 
forced by  600  men,  and  4  months  afterwards,  a  second  rein- 
forcement of  400  was  sent  in  :  how  long  would  the  provisions 
last  after  the  last  reinforcement  arrived  ? 

50.  A  merchant  bought  a  quantity  of  broadclotli  and  baize 
for  $488,80;  there  was  117^  yards  of  broadcloth,  at  $3,\  i)er 
yard;  for  every  5  yards  of  broadcloth  he  had  1^  yards  of 
baize  :  how  many  yards  of  baize  did  lie  buy,  and  what  did  it 
cost  him  per  yard  ? 


I'ROMTSCUOUS   EXAMPLES.  343 

^1.  If  the  freight  of  40  tierces  of  sugar,  aach  weighing  3i 
cui.,  for  150  miles,  costs  $42,  Avhat  must  be  paid  for  the  freiglit 
of  lOhhd.  each  weighing  12cwL,  for  50  miles  ? 

52.  If  1  pound  of  tea  be  equal  in  value  to  50  oranges,  and 
70  oranges  be  worth  84  lemons,  what  is  the  value  of  a  pound 
of  tea,  when  a  lemon  is  worth  2  cents  ? 

53.  What  amount  must  be  discounted,  at  7  per  cent,  to  make  a 
present  payment  of  a  note  of  $500,  due  2  years  8  months  hence  ? 

54.  If  the  interest  on  $225  for  4i  years  is$91,12i,  what 
would  be  the  interest  on  $640  at  the  same  rate  for  2i  years  ? 

55.  A  farmer  having  1000  bushels  of  wheat  to  sell,  can  have 
$1,75  a  bushel  cash,  or  $1,80  a  bushel  in  90  days  :  which  would 
be  most  advantageous  to  him,  money  being  worth  7  per  cent  ? 

56.  A  merchant  bought  goods  to  the  amount  of  $1575  on  9 
months  credit ;  he  sells  the  same  for  $1800  in  cash :  money 
being  worth  6  per  cent,  what  did  he  gain  ? 

57.  Three  persons  in  partnership  gain  $482,62  ;  A  put  in  f 
as  much  capital  as  B,  and  B  put  in  |  as  much  as  C :  what  was 
each  one's  share  of  the  gain  ? 

58.  A  father  divided  his  estate,  worth  $9268,60,  among  his 
4  children,  giving  A,  i  of  it,  B,  J,  and  C,  $5  as  often  as  he  gave 
D  $6  :  how  much  did  each  receive  ? 

59.  A  tax  of  $475,50  was  laid  upon  4  villages.  A,  B,  C,  and 
D  ;  it  was  so  distributed,  that  as  often  as  A  and  B  each  paid 
$5,  C  paid  87,  and  D,  $8  :  what  part  of  the  whole  tax  did  each 
villa2;e  nav? 

60.  There  are  1000  men  besieged  in  a  town,  with  provisions 
for  5  weeks,  allowing  each  man  16  ounces  a  day.  If  they  are 
reinforced  by  400  men,  and  no  relief  can  be  afforded  lill  the 
end  of  8  weeks,  what  must  be  the  daily  allowance  to  each  man  ? 

61.  A  reservoir  has  3  pipes,  the  first  can  fill  it  in  10  days, 
the  second,  in  16  days,  and  the  third  can  empty  it  in  '^0  days: 
in  what  time  will  the  cistern  be  filled  if  they  are  all  allowed  to 
run  at  the  same  time  ? 

62.  Two  persons,  A  and  B,  are  on  opposite  sides  of  a  ■wood, 
which  is  536  yards  in  cii'cumference  ;  they  begin  to  tra.vel  in 


'64:4:  ANALYSIS    AIS'D 

the  same  direction  at  the  same  time ;  A  goes  at  the  rate  ol"  11 
yards  a  minute,  and  B,  at  the  rate  of  34  yards  in  3  minutes : 
how  many  times  will  B  go  round  the  wood  before  luj  over- 
takes A  ? 

63.  Two  men  and  a  boy  were  engaged  to  do  a  piece  of  work  . 
one  of  the  men  could  do  it  in  10  days,  the  other  in  16  days,  and 
and  the  boy  could  do  it  in  20  days :  how  long  would  it  take 
them  to  do  it  together  ? 

64.  A  owes  B  $500,  of  which  $150  is  to  be  paid  in  3  months. 
$175  in  6  months,  and  the  remainder  in  8  months  :  Avhat  would 
be  the  equated  time  for  the  payment  of  the  whole  ? 

65.  If  42  men,  in  270  days,  working  8^  hours  a  day,  can 
build  a  wall  98-|  feet  long,  7^  feet  high,  and  2i  feet  thick  ;  in 
how  many  days  can  63  men  build  a  wall  451  feet  long,  6^  feet 
high,  and  31  feet  thick,  working  111  hours  a  day  ? 

GO.  After  one-third  part  of  a  cask  of  wine  had  leaked  away, 
21  gallons  were  drawn,  when  it  was  found  to  be  half  full :  how 
much  did  the  cask  hold  ? 

67.  A  man  had  a  bond  and  mort£raG;e  for  $2500,  dated  Julv 
1st,  1854.  He  is  not  satisfied  with  7  per  cent  annual  interest, 
and  on  the  first  day  of  September,  1854,  he  purchased  10  shares, 
of  $100  each,  of  railroad  stock,  at  115.  Nov.  1st,  he  bought 
8  shares  more  of  the  same  stock,  at  98;  and  on  April  1st, 
1855,  he  bought  5  shares  more  at  the  same  rate.  On  the  first 
days  of  August  and  February,  in  each  year,  he  received  a  regu- 
lar serai-annual  dividend  of  4  per  cent,  and  at  the  end  of  the 
year  (January  1st,  1856,)  sold  his  whole  stock  at  99  :  which 
was  the  more  profitable  investment,  and  how  much  ? 

68.  A  landlord  being  asked  how  much  he  received  for  (he 
rent  of  his  property,  answered,  tliut  after  deducting  9  cents  from 
each  dollar,  for  taxes  and  repairs,  there  remained  $3014,30  : 
wliat  was  the  amount  of  his  rents? 

69.  If  165  pounds  of  soap  cost  $16,50,  for  how  mucli  will  it 
be  nfcessary  to  sell  390  pounds,  in  order  to  gain  the  cost  of  3G 
pounds  ? 


PEOMISCUOUS    EXAMPLES.  345 

70.  What  is  tlie  height  of  a  wall  which  is  14^  yards  in  length, 
and  -j\  of  a  yard  in  thickness,  and  -which  has  cost  $406,  it  hav- 
ing been  paid  for  at  the  rate  of  $10  per  cubic  yard  ? 

71.  A  thief  escaping  from  an  officer,  has  40  miles  the  start, 
and  travels  at  the  rate  of  5  miles  an  hour ;  the  officer  in  pursuit 
travels  at  the  rate  of  7  miles  an  hour :  how  far  must  he  travel 
before  he  overtakes  the  thief  ? 

72.  Two  families  bought  a  barrel  of  flour  together,  for  which 
they  paid  $8,  and  agreed  that  each  child  should  count  half  as 
much  as  a  grown  person.  In  one  family  there  were  3  grown 
persons  and  3  children,  and  in  the  other,  4  grown  persons  and 
10  children ;  the  first  family  used  from  the  flour  2  wrecks,  and 
the  second  3  weeks :  how  much  ought  each  to  pay  ? 

73.  At  iii-'42  a  thousand,  how  much  lumber  should  be  given 
for  a  farm  containing  33A.,  2i?.,  16P.,  valued  at  $125  an  acre  ? 

74.  IIow^  many  building  lots,  each  50  feet  by  100  feet,  can  be 
made  out  of  2^  acres  of  ground  ? 

75.  A  person  pays  $150  for  an  insurance  on  goods,  at  3f  per 
cent,  and  finds  that  in  case  the  goods  are  lost,  he  will  receive 
the  value  of  the  goods,  the  premium  of  insurance,  and  $25  be- 
sides :  what  was  the  value  of  the  goods  ? 

76.  A  distiller  purchased  5000  bushels  of  rye,  which  he  can 
have  at  96  cents  a  bushel,  ready  money,  or  $1,  with  2  months' 
credit ;  Avhich  would  be  the  more  advantageous  to  him,  to  buy 
it  on  credit,  or  to  borrow  the  money  at  7  per  cent,  and  pay  the 
cash  ? 

77.  A  stockholder  bought  |  of  the  capital  of  a  company  at 
par  ;  he  sold  |-  of  his  purchase  at  par,  and  the  remainder  for 
125000,  and  by  the  latter  sale  made  $5000  :  what  was  the  value 
of  the  whole  capital  ? 

78.  How  many  bushels  of  grain  will  a  bin  contain,  that  is 
Sf{.  bin.  wide,  2ft.  Gin.  long,  and  Gft.  deep? 

79.  If  the  two  sides  of  a  triangle  arc  75  feet  and  90  feet, 
and  the  perpendicular  to  the  third  side  45  feet,  Avhat  is  the 
length  of  the  third  side  ? 

80.  Three  travellers  have  2160  miles  to  go  before  they  reach 


''46  ANALYSIS    AND 

the  end.  of  their  journey ;  the  first  goes   30  miles  a  day,  the 

second  27,  and  the  third  24:  how  many  days  should  one  set  out 
alter  another  that  they  may  arrive  together  ? 

81.  A  house  which  was  sold  a  second  time  for  §7180,  would 
have  given  a  profit  of  §420  if  the  second  projirietor  had  pur- 
chased it  $130  cheaper  than  he  did  :  at  what  price  did  he  pur- 
chase it  ? 

82.  A  piece  of  land  of  188  acres  was  cleared  by  two  com- 
panies of  men,  working  together ;  the  first  numbered  25  men, 
and  the  second  22  :  the  first  company  received  $84  more  than 
the  second  :  how  many  acres  did  each  company  clear,  and  what 
did  the  clearing  cost  per  acre  ? 

83.  I  have  three  notes  payable  as  follows  :  one  for  $100,  due 
Feb.  12th,  1856,  the  second  for  $400,  due  March  12th,  and  the 
third  for  $300,  due  April  1st:  what  is  the  average  time  of  pay- 
ment ? 

84.  How  many  marble  slabs,  152??.  square,  will  it  take  to 
pave  a  floor  32  feet  long,  and  25  feet  wide  ?  What  will  be  th^ 
cost  at  $3  a  square  yard  for  the  marble,  and  40  cents  a  square 
yard  for  labor  ? 

85.  A  man,  in  bis  will,  bequeathed  $500  to  A,  $425  to  B, 
$300  to  C,  $250  to  D,  and  $175  to  E  ;  but  after  settling  up  the 
estate  and  paying  expenses,  there  was  but  $1155  left:  what  is 
each  one's  share  ? 

86.  If  'dibs,  of  tea  are  worth  7lbs.  of  coffee,  and  lilbs.  of 
coffee  are  worth  ASlbs.  of  sugar,  and  ISlbs.  of  sugar  are  worth 
27 lbs.  of  soap  ;  how  many  pounds  of  soap  are  Gibs,  of  tea  worth  ? 

87.  "What  is  the  hour,  when  the  time  past  noon  is  ^  the  time 
to  midnight  ? 

88.  If  f  of  a  yard  of  cloth  cost  $|,  being  |-  of  a  yard  wide, 
what  is  the  value  of  f  of  a  yard  If  yards  wide,  of  the  same 
quality? 

89.  A  i'armer  sold  GO  fowls,  a  part  turkeys,  and  a  part  chick- 
ens ;  for  the  turkeys  he  received  $1,10  apiece,  and    for    tlio 
chickens  50  cents  apiece,  and  for  the  whole  he  received  $51,60 
how  many  were  there  of  each  ? 


PllOMISCUOUS   EXAMPLES.  847 

90.  A  person  hired  a  man  and  two  boys  ;  to  the  man  he  gave 
6  shiUings  a  day,  to  one  boy  4  shillings  luid  to  the  other  3  shil- 
lings a  day,  and  at  the  end  of  the  time  he  paid  them  104  shil- 
lings :  how  long  did  they  work  ? 

91.  Divide  $6471  among  3  persons,  so  tbat  as  often  as  the 
first  gets  $5,  the  second  will  get  $6,  and  the  third  $7. 

92.  Two  partners  have  invested  in  trade  $1G00,  by  which 
they  have  gained  $300 ;  the  gain  and  stock  of  the  second  amount 
to  $1140  :   what  is  the  stock  and  the  gain  of  each  "? 

93.  What  is  the  height  of  a  tower  that  casts  a  shadow  75.75 
yards  long,  at  the  same  time  that  a  perpendicular  staff  3  feet 
hiijh,  chives  a  shade  of  4.55  feet  in  length  ? 

94.  A  can  do  a  certain  piece  of  work  in  3  weeks  ;*  B  can  do 
3  times  as  much  in  8  weeks ;  and  C  can  do  5  times  as  much  in 
12  weeks :  in  what  time  can  they  all  together  do  the  lirst  piece 
of  work  ? 

95.  Tm'o  persons  pass  a  certain  point  at  an  inter\  al  of  4 
hours  ;  the  first  travelling  at  the  I'ate  of  1 1^,  and  the  second 
174t  miles  an  hour :  how  far  and  how  long  must  the  first  travel 
before  he  is  overtaken  by  the  second  ? 

96.  Three  persons  engage  in  trade,  and  the  sum  of  their 
stock  is  $1600.  A's  stock  was  in  trade  6  months,  B's  12  months, 
and  C's  15  months  ;  at  the  time  of  settlement,  A  receives  $120 
of  the  gain,  B  $400,  and  C  $100  :  what  was  each  person's 
stock  ? 

97.  A,  B  and  C,  start  at  the  same  time,  from  the  same  point, 
and  travel  in  the  same  direction,  around  an  island  73  miles  in 
circumference.  A  goes  at  the  rate  of  6  miles,  BIO  miles,  and 
C  16  miles  per  day:  in  what  time  will  they  all  be  together 
again  ? 

98.  What  length  of  wire,  |  of  an  inch  in  diameter,  can  be 
drawn  from  a  cube  of  copper,  of  2  feet  on  a  side,  allowing 
10  per  cent  for  waste  ? 

99.  A  person  having  $10000  invested  m  G  per  cent,  stocks, 
sells  out  at  65,  and  invests  the  proceeds  in  5  per  cents  at  82|^  • 
what  will  be  the  difference  in  his  income  ? 

100.  In   order  to  take  a  boat  through  a  lock  from  a   certain 


uiy  ANALYSIS    AND 

river  into  a  canal,  as  well  as  to  descend  from  the  canal  into  the 
river,  a  volume  of  water  is  necessary  461  yards  long,  8  yards 
wide  and  2|  yards  deep.  How  many  cubic  yards  of  water 
will  this  canal  throw  into  the  river  in  a  year,  if  40  boats  ascend 
and  40  descend  each  day  except  Sundays  and  eight  holidays  ? 

101.  A  company  numbering  sixty-six  shareholders  have  con- 
structed a  bridge  which  cost  $200000  :  what  will  be  the  gain 
of  each  partner  at  the  end  of  22  years,  supposing  that  6400 
persons  pass  each  day,  and  that  each  pays  one  cent  toll,  the 
expense  for  repairs,  &c.,  being  $o  per  year  for  each  share- 
holder ? 

102.  Five  merchants  were  in  partnership  for  four  years,  the 
first  put  in  $60,  then,  5  months  after.  $800  ;  the  second  put  in 
first  $600,  and  6  months  after  $1800  ;  tlie  third  put  in  $400; 
and  every  six  months  after  he  added  $500  ;  tlie  fourth  did  not 
contribute  till  8  months  after  the  commencement  of  the  part- 
nership ;  he  then  put  in  $900,  and  repeated  tliis  sum  every  6 
months  ;  the  fifth  put  in  no  capital,  but  kept  the  accounts,  for 
which  the  others  agreed  to  allow  him  $800  a  yeai-,  to  be  paid 
in  advance  and  put  in  as  capital.  What  is  each  one's  share  of 
the  gain,  which  was  $20,000  ? 

103.  A  general  arranging  his  army  in  the  form  of  a  square, 
finds  that  he  has  44  men  remaining,  but  by  increasing  each  side  by 
another  man,  he  wants  49  to  fill  up  the  square  :  how  many  men 
had  he  ? 

104.  A,  B  and  C,  are  to  share  $100  in  the  proportion  of 
1,  i  and  -},  respectively ;  but  by  the  death  of  C,  it  is  required 
to  divide  the  Avhole  sum  proportionally  between  the  other  two  : 
what  will  each  have  ? 

105.  A  lady  going  out  shopping  spent  at  the  first  place  she 
stopped,  one-half  her  money,  and  half  a  dollar  more ;  at  tho 
next  place,  half  tlie  remainder  and  half  a  dollar  more ;  and  at 
the  next  place  lialf  llio  remainder  and  half  a  dollar  more,  when 
she  found  lliat  she  liad  l)ut  three  dollars  left :  how  much  had 
she  when  she  started  ? 

100.  If  a  i)ipc  of  6  inches  discharge  a  certain  quantity  of 


PKOMISUUrOUS    EXAMPLES.  349 

fluid   in    4   hours,  in  "wliat    time  will  4  pipes,  each  of  3    inches 
bore,  discharge  twice  that  quantity  ? 

107.  A  man  bought  12  horses,  agreeing  to  pay  |40  for  the 
first,  and  in  an  increasing  arithmetical  progression  for  the  rest, 
paying  io70  foi'  the  last :  what  was  the  difference  in  the  cost, 
and  what  did  he  pay  for  them  all  ? 

108.  A,  B,  C  and  D,  engaging  in  speculation,  lost  a  sum  of 
money,  of  which  A,  B  and  C,  paid  $297,60  ;  B,  C  and  D, 
$321,92  ;  C,  D  and  A,  i5^375,83  ;  and  D,  A  and  B,  $402,50  • 
what  did  each  one  pay  ? 

109.  If  for  £3000  exchange  we  pay  7-|-  per  cent  premium, 
giving  in  payment  notes  at  4  months,  12  per  cent  discount,  what 
rate  ought  we  to  make  the  premium,  giving  notes  at  6  months, 
10  per  cent  discount  ? 

110.  A  purchase  of  $15000  worth  of  goods  is  to  be  paid  for 
in  three  equal  payments  without  interest ;  the  first  in  4  months, 
the  second  in  6  months,  and  the  third  in  9  months  :  money 
being  worth  7  per  cent,  how  much  ready  money  ought  to  pay 
the  debt  ? 

111.  If  an  iron  bar  5  feet  long,  2^-  inches  broad,  and  1-^- 
inches  thick,  weigh  45  pounds,  how  much  will  a  bar  of  the 
same  metal  weigh,  that  is  7  feet  long,  3  inches  broad,  and  2i 
inches  thick  ? 

112.  A  market  woman  bought  a  certain  number  of  eggs  at 
the  rate  of  4  for  3  cents,  and  sold  them  at  the  rate  of  5  for  4 
cents,  by  which  she  made  4  cents :  what  did  she  pay  apiece  for 
the  eggs  ?  What  did  she  make  on  each  egg  sold  ?  How  many 
did  she  sell  to  gain  4  cents  ? 

113.  A  person  passed  i  of  his  life  in  childhood,  yV  of  it  in 
youth,  5  years  more  than  -|-  of  it  in  matrimony :  he  then  had  a 
son,  whom  he  survived  4  years,  and  who  readied  only  J  the 
a2;e  of  his  father.     At  what  age  did  he  die  ? 

114.  A  well  is  to  be  stoned,  of  which  the  diameter  is-  6  feet  6 
inches,  the  thickness  of  the  wall  is  to  be  1  foot  G  inches,  leaving 
the  diameter  of  the   well   witliin   the  wall  3  feet  6  inches.     If 

10 


o50  PKOMISCUOUb    EXAMPLES. 

the  well  is  40  feet  deep,  how  many  cubic  feet  of  stone  will  be 

required  ? 

115,  A  surveyor  measured  a  piece  of  ground  in  the  form  of 
a  rectangle,  and  found  one  side  to  be  37  chains,  and  the  other 
42  chains  1 6  links  :  how  many  acres  did  it  contain  ? 

116,  A,  B  and  C,  can  buikl  a  barn  in  10  days;  after  4  days, 
A  leaves,  and  B  and  C  go  on  with  the  work  for  5  days  longer, 
when  B  leaves,  --^q  of  the  work  being  yet  unfinished  :  C  pro- 
ceeds with  the  work  and  finishes  it  in  11|-  days  after  B  left : 
how  long  would  it  take  each  to  build  the  barn  ? 

117,  A  tanner  bought  a  piece  of  land  for  ^1500,  and  agreed 
to  pay  principal  and  interest  in  5  equal  annual  instalments : 
if  the  interest  was  7  per  cent.,  how  much  was  the  annual  pay- 
ment ? 

118,  A  fountain  has  4  receiving  pipes,  A,  B,  C  and  D  ;  A, 
B  and  C  will  fill  it  in  G^- hours  ;  B,  C  and  Din  7\  hours  ;  C,  D 
and  A  in  8  hours ;  and  D,  A  and  B  in  15  Lours  :  it  has  also 
4  discharging  pipes,  E,  F,  G  and  H ;  E,  F  and  G  will  empty 
it  in 7^  hours  ;  F,  G  and  H  in  4  hours  ;  G,  II  and  E  ino]  hours  ; 
II,  E  and  F  in  31  hours.  Suppose  the  fountain  full  of  water, 
and  all  the  pipes  open,  in  what  time  would  it  be  emptied  ? 

119,  IIow  many  planks  15  feet  long,  and  15  inches  wide, 
will  floor  a  barn  GOJy  feet  long,  and  331  feet  wide  ? 

120,  If  a  ball  2  inches  in  diameter  weigh  5  pounds,  wjiat 
will  be  the  diameter  of  another  ball  of  the  same  material  that 
weighs  78,125  pounds  ? 

121,  A  gives  B  his  bond  for  $5000,  dated  April  1st,  1851, 
payable  in  10  equal  aniuuvl  instalments  of  $500  each,  on  and 
afier  the;  first  day  of  Ajjril.  1852.  A  afterwards  agreed  to 
take  up  his  bond  on  the  first  day  of  April,  1853,  deduding 
semi-annual  discount,  at  the  rate  of  7  per  oent,  per  annum,  on 
till-!  several  payments,  which  \'v\i  due  after  the  first  day  of 
Api-il,  i.sr>2:  what  sum,  on  the  first  day  of  April,  1853,  will 
o*UH',':l  the  bond  ? 


APPLICATIONS  OF  AllITHMETIC. 


MENSURATION. 

329.  Mensukation  embraces  all  the  methods  of  determining 
the  contents  of  geometrical  figuies.  It  is  divided  into  two  parts, 
the  raeniiiration  of  surfaces  and  the  mensuration  of  Volumes. 


1  foot. 


o 


MENSURATION    OP    SURFACES. 

330.   Surfaces  have  length  and  breadth.     They  are  measured 
by  means  of  a  square,  which  is  called  the  unit  of  surface. 

A  square  is  the  space  included  between  four 
equal  lines,  drawn  perpendicular  to  each  other. 
Each  line  is  called  a  side  of  the  square.  If 
each  side  be  one  foot,  the  figure  is  called  a 
square  foot. 

If  the   sides  of  a  square  be  each  four  feet, 
the  square  will  contain   sixteen  square  feet.     For,  in  the  large 
square  there  are  sixteen  small  squares,  the  sides  of  which  are 
each  one  foot.     Thei'efore,  the  square  whose  side  is  four  feet, 
contains  sixteen  square  feet. 

The  number  of  small  squares  that  is  con- 
tained in  any  large  square  is  always  equal  to 
the  product  of  two  of  the  sides  of  the  large 
square.  As  in  the  figure,  3x3  =  9  square 
feet.  The  number  of  square  inches  contained 
in  a  square  foot  is  equal  lo  12  x  12  =  144. 


329.  What  is  mensuration  1 

330.  What  is  a  surface  i     What  is  a  square  1     ^Miat  is  tlic  luiiuber  ol 
small  fciiuart's  coutained  ia  a  large  square  equal  to  1 


MElvrSUKATION 


331.  A  triangle  is  a  figure  bounded  by  three  straight  lines. 
Thus,  ACB  is  a  triangle. 

The  lines  BA,  AC,  BC,  are  called  sides  ; 
and  the  corners,  B,  A  and  C,  are  called 
angles.     The  side  AB  is  the  base. 

When  a  line  like  CD  is  drawn,  making 
the  angle  CDA  equal  to  the  angle  CDB, 
then  CD  is  said  to  be  perpendicular  to  AB,  and  CD  is  called 
the  altitude  of  the  triangle.  Each  triangle  CAD  or  CDB  is 
called  a  right-angled  triangle.  The  side  BC,  or  the  side  AC, 
opposite  the  right  angle,  is  called  the  hypothenuse. 

The  area  or  contents  of  a  triangle  is  equal  to  half  the  product 
of  its  base  by  its  altitude  (Bk.  IV.,  Prop.  VI).* 


OPERATION. 

50 
30 


2)1500 
Ans.     1  bO  square  yards. 


EXAMPLES. 

1.  The  base,  AB,  of  a  triangle  is 
50  yards,  and  the  perpendicular,  CD, 
80  yards  :  what  is  the  area  ? 

Analysis. — We  first  multiply  the  base 
by  the  altitude,  and  the  product  is  square 
yards,  which  "vve  divide  by  2  for  the  area. 

2.  In  a  triangular  field  the  base  is  GO  chains,  and  the  pei*- 
pendicular  12  chains  :  how  much  does  it  contain  ? 

3.  There  is  a  triangular  field,  of  wliich  the  base  is  45  rods, 
and  tlic  perpendicular  38  rods :  what  are  its  contents  ? 

4.  Wliat  are   the  contents   of  a  triangle  whos.e  base  is  75 
chains,  and  perpendicular  36  chains  ? 

332.  A  rectangle  is  a  four-sided  figure  like 

a  squai'e,  in  which  the  sides  are  perpendicular 

to  each  other,  but  the  adjacent  sides  are  not 

?qual. 

*  All  the  references  are  to  Davies'  Letrendrc. 


331.  What  is  a  triangle  ?  What  is  the  base  of  a  triangle'!  What  the 
I'lliliulo?  What  is  a  rirjht-angled  triangle  1  Which  side  is  the  hypo- 
tlicniise  1     What  is  the  aica  of  a  triangle  equal  to  \ 

■M'i.  Wluit  i^  a  rectangle  ? 


\ 


or  suKFACKs.  353 

333.  A  parallelogram  is  a  foux*-sided 
figure  which  has  its  opposite  sides  equal 
and  parallel,  but    its    angles    not    right- 
angles.     The  line  DE,  perpendicular  to      -    p 
the  base,  is  called  the  altitude. 

334.  To  find  the  area  of  a  square,  rectangle,  or  parallelogram. 

Multiply  the  hase  by  the  2^cfpendicular  height,  and  the  -product 
will  he  the  area  (Bk.  IV.,  Prop.  Y.) 

EXAMPLES. 

1.  What  is  the  area  of  a  square  field,  of  which  the  sides  are 
each  6G.16  chains  ? 

2.  What  is  the  area  of  a  square  piece  of  land,  of  which  the 
sides  are  54  chains  ? 

3.  What  is  the  area  of  a  square  piece  of  land,  of  which  the 
sides  are  75  rods  each  ?  ' 

4.  What  are   the  contents  of  a  rectangular  field,  the  leno;th 
of  which  is  80  rods,  and  the  breadth  40  rods  ? 

5.  What  are  the  contents  of  a  field  80  rods  square  ? 

6.  What  are  the  contents  of  a  rectangular  field,  30  chains 
long  and  5  chains  broad  ? 

7.  What  are  the  contents  of  a  field,  54  chains  long  and  18 
rods  broad  ? 

8.  The  base  of  a  parallelogram  is  542  yards,  and  the  per- 
pendicular height  720  feet :  Avhat  is  the  area  ? 

335.  A  trapezoid  is  a  four-sided  figure        d         E 
ABCD,  having  two  of  its  sides,  AB,  DC, 
parallel.     The  perpendicular  EF  is  call- 


-^d  the  altitude.  A  F  B 

336.  To  find  the  area  of  a  trapezoid. 

Multiply  the  sicni  of  the  two  parallel  sides  hy  the    altitude, 

333.  What  is  a  parallelogram  T 

834.  How  do  you  find  the  area  of  a  square,  rectangle,  or  parallelogram  1 
335.  ^^'hat  is  a  trapezoid  1 
330.  How  do  you  find  the  area  of  a  trapezoid  ? 

If) 


854: 


MENSURATION 


OPERATION. 
643.02  +  428.48  =  1071.50  = 
sum  of  parallel  sides.  Then, 
1071.50x342.32  =  366795.88; 
and  S^&l^&JSA  ^  183397.94  = 
the  area. 


divide  the  p7-oduct  hj  2,  and  the  quotient  will  he  the  area  (Bk. 
IV.  Prop.  VII). 

EXAMPLES. 

1.  Required  the  area  or  contents  of  the  trapezoid  ABCD,  hav- 
ing given  AB=G43.02  feet,  DC  = 

428.48  feet,  and  EF  =  342.32  feet. 

Analysis. — We  tirst  find  the  sum 
of  the  sides,  and  then  multiply  it  by 
the  perpendicular  height,  after  which 
we  divide  the  product  by  2.  for  the 
area. 

2.  What  is  the  area  of  a  trapezoid,  the  parallel  sides  of  which 
are  24.82  and  16.44  chains,  and  the  perpendicular  distance 
between  them  10.30  chains? 

3.  Required  the  area  of  a  trapezoid,  whose  parallel  sides  are 
51  feet  and  37  feet  6  inches,  and  the  perpendicular  distance 
between  them  20  feet  and  10  inches. 

4.  Required  the  area  of  a  trapezoid,  whose  parallel  sides  are 
41  and  24.5,  and  the  perpendicular  distance  between  them  21.5 
yards. 

5.  What  is  the  area  of  a  trapezoid,  whose  parallel  sides  are 
15  chains,  and  24.5  chains,  and  the  perpendicular  height  30.80 
chains  ? 

6.  What  are  the  contents  of  a  trapezoid,  when  the  parallel 
sides  are  40  and  64  chains,  and  the  perpendicular  distance 
between  them  52  chains  ? 

337.  A  circle  is  a  portion  of  a  plane  bounded  by  a  curved 
line,  every  point  of  which  is  equally  dis- 
tant from  a  certain  point  within,  called 
the  centre. 

The  curved  line  AEBD  is  called  the 
circumference;  the  point  C  the  centre  ; 
the  line  AB  passing  through  the  centre 
Ti,  did  meter ;  and  C'li  n  radius. 

TIic  circiiniierence  AEBD  is  3.1416 
times    as    great   as    the    diiuneter   AB- 


OF   SUKFACES,  855 

Hence,  if  the  diameter  is  1,  the  circumference  Avill  be  3.1416. 
Therefore,  if  tlie  diameter  is  known,  the  circumference  is  found 
hy  multiplying  3.1416  by  the  diameter  (Bk.  V.  Prop.  XIV). 

EXA3IPLES.  ' 

1.  The  diameter  of  a  circle  is  8  :  what  is  the  circumference  ? 

OPERATION. 

Analysis. — The  circumference  is  found  3.1416 

by  simply  multiplying  3.1416  by  the  di-  8 

ameter.  »  Ans.  25.1328 

2.  The  diameter  of  a  circle  is  186  :  what  is  the  circum- 
ference ? 

3.  The  diameter  of  a  circle  is  40  :  what  is  the  circum- 
ference ? 

4.  What  is  the  circumference  of  a  circle  whose  diameter  is 

? 

338.  Since  the  circumference  of  a  circle  is  3.1416  times  as 
great  as  the  diameter,  it  follows,  that  if  the  circumference  is 
known,  we  may  find  the  diameter  by  dividing  it  by  3.1416. 

EXABIPLES. 

1.  What  is  the  diameter  of  a  circle  whose  circumference  is 
157.08? 

OPERATION. 

Analysis. — We  divide  the  circumference        3.1416)157.080(50 
by  3.1416,  the  quotient  50  is  the  diameter.  157.080 

2.  Wliat  is  the  diameter  of  a  circle  whose  circumference  i3 
23304.3888  ? 

3.  What  is  the  diameter  of  a  circle  whose  circumference  is 
13700  ? 

337.  What  ir-  a  circle  1  What  is  the  centre  ?  What  is  the  circumfer- 
ence ':  What  is  the  diameter  1  What  the  radius  1  How  many  times 
greater  is  tliC  circumference  th.m  the  diameter  1  How  do  you  find  the 
circumference  when  tlie  diameter  is  known  ? 

33S.  How  do  you  find  the  diameter  when  the  circumference  is  knuwu  1 


356  MEKSdRATION 

339.  To  find  the  area  or  contents  of  a  circle. 

Multlphj  the  square  of  the  diameter  by  the  decimal  .7854 
(Bk.  V.  Prop.  XII.  Cor.  2). 

EXA3IPLES. 

1.  What  is  the  area  of  a  circle  whose  diameter  is  12  ? 


Analysis.— We  first  square  the  diam-  operation. 

2 

eter,  giving  144,  which  we  then  multi-  12  =  144 

ply  by  the  decimal  .7854  :  the  product     144  x  .7854  =  113.0976 

is  the  area  of  the  circle.  Ans.  113.0976 


2.  What  is  the  area  of  a  circle  whose  diameter  is  5  ? 

3.  What  is  the  area  of  a  circle  whose  diameter  is  14  ? 

4.  How  many  square  yards  in  a  circle  whose  diameter  is  d^ 
feet? 

340.  A  sphere  is  a  portion  of  space 

bounded  by  a  curved   surface,  all  the 

points  of  which  are  equally  distant  from 

a  certain  point  within,  called  the  centre. 

The  line  AD,  passing  through  its  centi-e 

C,  is  called  the  diameter  of  the  sphere, 

and  AC  its  radius. 

A 

341.  To  find  the  surface  of  a  sphere, 

Multiply  the  square  of  the  diameter  by  3.1416   (Bk.  VHL 
Prop.  X.  Cor.) 

EXAMPLES. 

1.  What  is  the  surface  of  a  sphere  whose  diameter  is  6  ? 

OPERATION. 

Analysis. — We  simply  multiply  the  number  3,1416 

3.1416  by  the  .square  of  the  diameter  :  the  pro-     6'=  36 

duct  is  the  surface.  Ans.   1 13.076 


339.  How  do  you  find  the  area  of  a  circle  ? 

340.  What  is  a  sphrre  ?     \\'hat  is  a  iliameterl     What  is  a  radius^ 

341.  How  do  you  find  the  surface  of  a  gpherc  ? 


OF   VOLUME. 


357 


3  feet  =  1   yard. 


2.  What  is  the  surface  of  a  sphere  whose  diameter  is  1 4  ? 

3.  Required  the  number  of  square  inches  in  the  surface  of  a 
sphere  whose  diameter  is  3  feet  or  36  inches. 

4.  Required  the  area  of  the  surface  of  the  earth,  its  mean 
diameter  being  7918.7  miles. 

MENSURATION  OF  VOLUMES. 

342.  A  SOLID  or  VOLUME  is  a  portion  of  space  h-iving  tliree 
dimensions  :  length,  breadth,  and 

thickness.  It  is  measured  bj  a 
cule  called  the  cuhic  unit, or  unit 
of  vohime. 

A  CUBE  is  a  volume  having  six 
equal  faces,  which  are  squares.  If 
the  sides  of  the  cube  lie  each  one 
foot  long,  the  figure  is  called  a 
cubic  foot.  But  when  the  sides 
of  the  cube  are.  one  yard,  as  in  the 

figure,  it  is  called  a  cubic  yard.  The  base  of  the  cube,  which 
is  the  face  on  which  it  stands,  contains  3x3  =  9  square  feet. 
Therefore,  9  cubes,  of  one  foot  each,  can  be  placed  on  the  base. 
If  the  figure  were  one  foot  high  it  would  contain  9  cubic  feet ; 
if  it  were  2  feet  high  it  would  contain  two  tiers  of  cubes,  or  18 
cubic  feet ;  and  if  it  were  3  feet  high,  it  would  contain  three 
tiers,  or  27  cubic  feet.  Hence,  the  contents  of  stick  a  f  (jure  arc 
equal  to  the  "product  of  its  lengthy  breadth,  and  height. 

343.  To  find  the  contents  of  a  sphere, 

Multiply  the  surface  by  the  diameter,  and  divide  the  product 
by  6,  the  quotient  will  be  the  contents  (Bk.  VIII.  Prop.  XIV. 
Sch.  3). 

EXAMPLES. 

1.  What  are  the  contents  of  a  sphere  whose  diameter  is  12  ? 

342.  What  i.s  a  volume  ]  AVhat  is  a  cube  1  W!i«t  is  a  cubic  foot  ? 
What  is  a  cubic  yard  1  How  many  cubic  feet  in  a  cubic  yard  ■!  What  are 
the  contents  of  a  figure  of  three  dimensions  equal  to  1 

:i'iri.   Hov.'  di>  you  ihid  the  contetit.s  of  a  sphered 


858 


MENSUKATION 


Analysis. — We  first  find  the  surface 
by  multiplying  the  square  of  the  diam- 
eter by  3.1416.  We  then  multiply  the 
surface  by  the  diameter,  and  divide  the 
product  by  6. 


OPi  RATION, 

12"=  144 
mulliply  by  3.1416 
surface  452.3904 

diameter  ]  2 

solidity 


904.7808 


2.  What  are  the  contents  of  a,  sphere  whose  diameter  is  8  ? 

3.  What  are  the  contents  of  a  sphere  whose  diameter  is  16 
inches  ? 

4.  What  are  the  contents  of  the  earth,  its  mean  diameter 
being  7918.7  miles? 

5.  What  are  the  contents  of  a  sphere  whose  diameter  is  12 
feet? 

344.  A  prism  is  a  volume  whose  ends  or  bases 
are  equal  plane  figures  and  whose  faces  are  par- 
allelograms. 

The  sum  of  the  sides  which  bound  the  base  is 
called  the  perimeter  of  the  base,  and  the  sum  of 
the  pai'allelograms  which  bound  the  prism  is  called 
the  convex  surface. 

34.5.  To  find  the  convex  surface  of  a  ri"'ht 
prism. 

Multiply  the  perimeter  of  the  base  hi/  the  perpendicular 
height,  and  the  product  will  he  the  convex  surface  (Bk.  VII. 
Prop.  I). 

EXAMPLES. 

1.  What  is  the  convex  surface  of  a  prism  whose  base  is 
bounded  by  five  equal  sides,  each  of  which  is  35  feet,  the  alti- 
tude being  52  feet  ? 

2.  AVhat  is  the  convex  surface  when  there  are  eight  equal 
sides,  each  15  feet  in  length,  and  tlie  altitude  is  12  feet? 


344.  What  is  a  prism  1     What  is  the  perimeter  of  the  base  1     What  ia 
the  convex  surface  ] 

345.  How  do  you  find  tlic  convex  surface  of  a  prism  ? 
34G.  How  do  you  find  Iho  contPiit:!  of  a  prinui  1 


OF    VOLUM]':. 


;J59 


346.  To  find  the  contents  of  a  prism, 

Multiply  the  area  of  the  base  by  the  altitude,  and  the  product 
will  be  the  contents  (Bk.  VII.  Prop.  XIV). 

EXAMPLES. 

1.  What  are  the  contents  of  a  square  prism,  each  side  of  the 
square  -which  forms  the  base  being  IG,  and  the  ahitude  of  the 
prism  30  feet  ? 

OPERATION. 

Analysis. — We  first  tind  the  area  of  the  square  16^—  256 

which  forms  the  base,  and  then  multiply  by  the  30 

altitude.  Ans.  7680 

2.  What  are  the  contents  of  a  cube,  each  side  of  Avliich  is 
48  inches  ? 

3.  How  many  cubic  feet  in  a  block  of  marble,  of  which  the 
length  is  3  feet  2  inches,  breadth  2  feet  8  inches,  and  height,  or 
thickness  5  feet  ? 

4.  How  many  gallons  of  water  will  a  cistern  contain,  whose 
dimensions  are  the  same  as  in  the  last  example  ? 

5.  Required  the  solidity  of  a  triangular  prism,  whose  height 
is  20  feet,  and  area  of  the  base  G91. 

347.  A  Cylinder  is  a  volume  generated 
by  the  revolution  of  a  rectangle  AF  about 
EF.  The  line  EF  is  called  the  axis  or  alti- 
tude— the  circular  surface,  the  convex  surface 
of  the  cylinder,  and  tlie  circular  ends,  the  bases. 

348.  To  find   the    convex   surface  of  a 
cylinder, 

Multiply  the  circumference  of  the  base  by  the  altitude,  and  the 
product  will  be  the  convex  surface  (Bk.  VIII.  Prop.  I). 


347.  What  is  a  cylinder  1     What  is  the  axis  or  altitude  T     What  is  tho 
convex  surface  1 

348.  How  do  you  find  the  convex  surface  ! 


3  GO  MENSTIRA.TION 

EXASIPLES. 

1.  "What  is  the  convex  surface  of  a  cylinder,  the  diameter 
of  whose  base  is  20  and  the  altitude  40  ? 

OPKRATION. 

AxALVSis. — We  first  multiply  3.1416  by  3.1416 

the  diameter,  which  gives  the  circumfer-  20 

ence  of  the  base.      Then,  multiplying  by  62.8320 

the   altitude,  we   obtam   the   convex  sur-  40 

face.  Ans.  2513.2800 


2.  "What  is  the  convex  surface  of  a  cylinder  whose  altitude 
is  28  feet  and  the  circumference  of  its  base  8  feet  4  inches  ? 

3.  What  is  the  convex  surface  of  a  cylinder,  the  diameter 
of  whose  base  is  15  inches  and  altitude  5  feet  ? 

4.  What  is  the  convex  surface  of  a  cylinder,  the  diameter 
of  whose  base  is  40  and  altitude  50  feet  ? 

349.  To  find  the  volume  of  a  cylinder, 

3Ii(ltiph/  the  area  of  the  base  by  the  altitude :  the  product  will 
he  the  contents  or  volume  (Bk.  VIII.  Prop.  II), 

EXAMPLES. 

1.  Required  the  contents  of  a  cylinder  of  which  the  altitude 
is  11  feet,  and  the  diameter  of  the  base  16  feet. 

Analysis. — We  first  find  the  area  of  the  operation. 

base,  and  then  multiply  by  the  altitude:  16^  =  256 

the  product  is  the  solidity.  .7854 

2.  What  are  the  contents  of  a  cylin-         ^"^^^  ^^'®'  201-0624 

der,  the  diameter  of  whose  base  is  40,  

'  ,  .     ,     „^^  2211.6864 

and  the  altitude  29  ? 

3.  ^Vliat  are  the  contents  of  a  cylinder,  the  diameter  of  whose 
base  is  24,  and  the  altitude  30  ? 

4.  "V\''hat  are  the  contents  of  a  cylinder,  the  diameter  of  whose 
base  is  32,  and  altitude  12  ? 

5.  "What  are  the  contents  of  a  cylinder,  the  diameter  of  whose 
base  is  25  feet,  and  altitude  15  ? 

U4'J    IIow  do  you  find  the  contents  of  a  cyliudcr  1 


OF   VOLTTME. 


861 


350.  A  Pyramid  is  a  volume  bounded 
by  several  triangular  planes  united  at  the 
same  point  S,  and  by  a  plane  figure  or  base 
ABODE,  in  which  they  terminate.  The 
altiude  of  the  pyramid  is  the  line  SO, 
drawn  perpendicular  to  the  base. 


351.  To  find  the  contents  of  a  pyramid. 

3fultipli/  the  area  of  the  base  by  the  altitude,  and  divide  thd 
product  by  3  (Bk.  VII.,  Prop.  XVII). 

EXAMPLES. 

1.  Required  the  contents  of  a  pyramid,  the 
area  of  whose  base  is  86,  and  the  altitude  24. 


OPERATION. 

86 

24 

3);i0tJ4 


Analysis. — We  simply  multiply  the  area  of  the 
base  86,  by  the  altitude  24.  and  then  divide  the  Ans.     688 

product  by  3. 

2.  What  are  the  contents  of  a  pyramid,  the  area  of  whose  baso 
is  3G5,  and  the  altitude  36  ? 

3.  What  are  the  contents  of  a  pyramid,  the  area  of  whose  base 
is  207,  and  altitude  36  ? 

4.  What  are  the  contents  ef  a  pyramid,  the  area  of  whose 
base  i.s  oG2,  and  altitude  30  ? 

5.  What  are  the  contents  of  a  pyramid,  the  area  of  whose 
base  is  540,  and  altitude  32  ? 

6.  A  pyramid  has  a  rectangular  base,  the  sides  of  which  are 
50  and  24;  the  altitude  of  the  pyramid  is  36:  what  are  its 
contents  ? 

7.  A  pyramid  with  a  square  base,  of  which  each  side  is  15, 
has  an  ahitude  of  24:  what  are  its  contents  ? 

350.  What  is  a  pyramid  1     What  is  the  altitude  of  a  pyramid  ] 

351.  How  do  you  find  the  contents  of  a  pyramid  1 


GAUGING. 


352.  A  Cone  is  a  volume  generated  by 
the  revolution  of  a  ri^'bt  ani^led  trianerle 
ABC,  about  the  side  CB.  The  point  C  is 
tli(!  vertex,  and  the  line  CB  is  called  the 
axis  or  altitude. 


353.  To  find  the  contents  of  a  cone. 

Multiply  the  area  of  the  base  hj  the  altitude,  and  divide  the 
product  by  3  ;  or,  midtiply  the  area  of  the  base  by  one-third  of 
the  altitude  (Bk.  VIII.,  Prop.  V.) 

EXAMPLES. 

1.  Required  the  contents  of  a  cone,  the  diameter  of  whose 


base  is  6,  and  the  altitude  11. 


OPERATION. 

6^  =  36 

.7854  =  28.2744 

n 

3)311.0184 
Ans.      103.(j728 


Analysls. — We  fir.st  square  the  diame- 
ter, and  multiply  it  by  .7854,  which  gives     36  x 
the  area  of  the  base.     We  next  multiply 
by  the  altitude,  and  then   divide  the  pro- 
duct by  3. 

2.  What  are  the  contents  of  a  cone,  the  diameter  of  whose 
base  is  36,  and  the  altitude  27  ? 

3.  What  are  the  contents  of  a  cone,  the  diameter  of  whose 
base  is  35,  and  the  altitude  27  ? 

4.  What  are  the  contents  of  a  cone,  whose  altitude  is  27  feet, 
and  the  diameter  of  the  base  20  feet  ? 


GAUGING. 
354.   Cask-Guaging  is  the  method  of  finding  the  number 
of  gallons  which  a  cask  contains,  by  measuring  the  external 
dimensions  of  the  cask. 

352.  Wh&t  is  a  cone  \     What  is  the  vertex  ?     What  is  the  axis  I 
3.')3.   How  do  you  find  llio  contents  of  a  cone  ? 
364.    What  is  cask-gauging  \ 


GAUGING. 


S63 


355.  Casks  are  divided  into  four  varieties,  according  to  the 
curvature  of  their  sides.  To  wliicli  of  the  varieties  any  cask 
belongs,  must  be  judged  of  by  inspection. 


1st  Variety — least  curvature. 


2d  Variety. 


3d  Variety. 


4th  Variety — greatest  curvature. 


356.  The  first  thing  to  be  done  is  to  find  the  mean  diameter. 
To  do  this. 

Divide  the  head  diameter  hy  the  hung  diameter,  and  find  the 
quotient  in  the  first  column  of  the  following  table,  marked  Qu. 
Then  if  the  hung  diameter  he  midtiplied  hy  the  number  on  the 
same  line  with  it,  and  in  the  column  answering  to  the  proper 
variety,  the  product  will  he  the  true  mean  diameter,  or  the  diame 
tcr  of  a  cylinder  having  the  same  altitude  and  the  same  con 
tents  with  the  cask  projMsed. 


oCi5.  Into  how  many  varieties  are  casks  divided  ! 
^56.  How  do  you  liiid  tlic  mean  diameter  1 


3(54 

GAUGING. 

Qu. 
50 

mVai. 

•2dVar. 

3dVar.  4thVar. 

Qu. 

1st  Var. 

2d  Var. 

3d  Var. 

4th  Var. 

8660 

8465 

7905 

7637 

76 

9270 

9227 

8881 

8827 

51 

8680 

8493 

7937 

7681 

77 

9296 

9258 

8944 

8874 

52 

8700 

8520 

7970 

7725 

78 

9324 

9290 

8967 

8922 

53 

8720 

8548 

8002 

7769 

79 

9352 

9320 

9011 

8970 

54 

8740 

8576 

8036 

7813 

80 

9380 

9352 

9055 

9018 

55 

8760 

8605 

8070 

7858 

81 

9409 

9383 

9100 

9066 

56 

8781 

8633 

8104 

7902 

82 

9438 

9415 

9144 

9114 

57 

8802 

8662 

8140 

7947 

83 

9467 

9446 

9189 

9163 

58 

8824 

8690 

8174 

7992 

84 

9496 

9478 

9234 

9211 

59 

8846 

8720 

8210 

8037 

85 

9526 

9510 

9280 

9260 

60 

8869 

8748 

8246 

8082 

86 

9556 

9542 

9326 

9308 

61 

8892 

8777 

8282 

8128 

87 

9586 

9574 

9372 

9357 

62 

8915 

8806 

8320 

8173 

88 

9616 

9606 

9419 

9406 

63 

8938 

8835 

8357 

8220 

89 

9647 

9638 

9466 

9455 

64 

8962 

8865 

8395 

8265 

90 

9678 

9671 

9513 

9504 

65 

8986 

8894 

8433 

8311 

91 

9710 

9703 

9560 

9553 

66 

9010 

8924 

8472 

8357 

92 

9740 

9736 

9608 

9602 

67 

9034 

8954 

8511 

8404 

93 

9772 

9768 

9656 

9652 

68 

9060 

8983 

8551 

8450 

94 

9804 

9801 

9704 

9701 

69 

9084 

9013 

8590 

8497 

95 

9836 

9834 

9753 

9751 

70 

9110 

9044 

8631 

8544 

96 

9868 

9867 

9802 

9800 

71 

9136 

9074 

8672 

8590 

97 

9901 

9900 

9851 

9850 

72 

9162 

9104 

8713 

8637 

98 

9933 

9933 

9900 

9900 

73 

91 88 

9135 

8754 

8685 

99 

9966 

9966 

9950 

9950 

74 

9215 

9166 

8796 

8732 

100 

10000 

10000 

10000 

10000 

75 

9242 

9196 

8838 

8780 

EXAMPLES. 

1.  Supposing  the  diameters  to  be  32  and  24,  it  is  required  to 
find  the  mean  diameter  for  each  variety. 

Dividing  24  by  32,  we  obtain  .75  ;  which  being  foimd  in  the 
column  of  quotients,  opposite  thereto  stand  the  numbers. 


.9242 
.919G 
.8838 
.8780 


which  being  each  mul- 
>.  tiplied  by  82,  produce 
respectively, 


20.5744 

29.4272 
28.2816 
28.09G0 


for  the  correspond- 
\.  ing  menn  diameters 
required. 


2.  The  head  diameter  of  a  cask  is  2G  inches,  and  the  bung 
diameter  3  feet  2  inches :  what  is  the  mean  diameter,  the  cask 
being  of  the  thii'd  variety  ? 

3.  The  head  diameter  is  22  inches,  the.  bung  diameter  34 
inches  ••  what  is  the  mean  diameter  of  a  cask  of  the  fourth  variety? 


GAUGING.  SG5 

357.  Having  found  the  mean  diameter,  we  multiply  the  square 
of  the  mean  diameter  by  the  decimal  .7854,  and  the  product  by 
the  length  ;  this  -will  give  the  contents  in  cubic  inches.  Then, 
if  we  divide  by  231,  we  have  the  contents  in  wine  gallons  (see 
Art.  414),  or  if  we  divide  by  282,  we  have  the  contents  in  beer 
gallons  (Art.  415). 

Analysis. — For  wine  measure,  we  mul-  operation. 

tiply  the  length  by  the  square  of  the  mean  I  X  d^  x   tj^^*  = 

diameter,  then  by  the  decimal  .7854,  and  I  X  d^  X  .0034. 
divide  by  231. 

If  then,  we  divide  the  decimal  .7854  by  231,  the  quotient  carried 
to  four  places  of  decimals  is  .003  1,  and  this  decimal  multiplied  by 
the  square  of  the  mean  diameter  and  by  the  length  of  the  cask,  will 
give  the  contents  in  wine  gallons. 

For  similar  reasons,  the  content  is  found  operation. 

in  beer  gallons  by  multiplying  together  the  I  X  d-  X  'l^^  = 

length;  the  square  of  the  mean  diameter,         I  X  d'-  X  .0028. 
and  the  decimal  .0028. 

Hence,  for  gauging  or  measuring  casks, 

31ultiply  the  length  by  the  square  of  the  mean  diameter  ;  then 
midt'ply  by  34  for  wine,  and  by  28  for  beer  measure,  and  point 
off  in  the  product  four  decimal  places.  The  product  will  then 
express  gallons  and  the  decimals  of  a  gallon. 

1.  How  many  wine  gallons  in  a  cask,  whose  bung  diameter 
is  36  inches,  head  diameter  30  inches,  and  length  50  inches; 
the  cask  being  of  the  first  variety  ? 

2.  What  is  the  number  of  beer  gallons  in  the  last  example  ? 

3.  How  many  wine,  and  how  many  beer  gallons  in  a  cask 
whose  length  is  36  inches,  bung  diameter  35  inches,  und  head 
diameter  30  inches,  it  being  of  the  first  variety  ? 

4.  How  many  wine  gallons  in  a  cask  of  which  the  head 
diameter  is  24  inches,  bung  diameter  36  inches,  and  length 
3  feet  6  inches,  the  cask  being  of  the  second  variety  ? 

t357.  How  do  you  find  the  contents  in  cubic  inches  1     How  do  you  find 
the  contents  in  wine  gallons  1     lu  beer  gallons  1 


S6(J  MECHANICAL   P0WEK3. 


OF  THE  MECHANICAL  POWERS. 

358.  There  are  six  simple  machines,  which  are  called 
Mechanical  2)0wers.  They  are,  the  Lever,  the  Pulley,  the 
Wheel  and  Axle,  the  Inclined  Plane,  the  Wedge,  and  the  Screw. 

359.  To  understand  the  nature  of  a  machine,  four  things 
must  be  considered. 

1st.  The  power  or  force  which  acts.  This  consists  in  the 
efforts  of  men  or  horses,  of  weights,  springs,  steam,  &;c. 

2d.  The  resistance  which  is  to  be  overcome  by  the  power. 
This  generally  is  a  weight  to  be  moved. 

3d.  We  are  to  consider  the  centre  of  motion,  or  fidcrum, 
which  means  a  prop.  The  prop  or  fulcrum  is  the  point  about 
which  all  the  parts  of  the  machine  move. 

4th.  We  are  to  consider  the  respective  velocities  of  the  power 
and  resistance. 

360.  A  machine  is  said  to  be  in  equilibrium  when  the  resist- 
ance exactly  balances  the  power,  in  which  case  all  the  parts  of 
the  machine  are  at  rest,  or  in  uniform  motion. 

We  shall  first  examine  the  lever. 

361.  The  Lever,  is  a  bar  of  wood  or  metal,  which  moves 
around  a  fixed  point,  called  the  fulcrum.  There  are  three 
kinds  of  levers. 


1st.  When  the  fulcrum  is 
between  the  weight  and  the 
power. 


358.  How  many  simple  machines  are  there  ^     What  are  they  called  1 

359.  What  things  must  be  considered,  in  order  to  understand  the  power 
of  a  machine  1 

300.   When  is  a  machine  said  to  be  in  equilibrium? 

361.  What  is  a  lever  1  How  many  kinds  of  levers  are  there  ?  Describe 
the  first  kind  \  Where  is  the  weight  placed  in  the  second  kind  \  Where 
ia  the  power  placed  in  the  third  kind  ] 


MECIIANIUAL   POWERS.  367 


2d.  "When  the  weight  is  be- 
tween the  power  and  the  ful- 
crum. 


3d.  When  the  power  is  be- 
tween the  fulcrum  and  the 
weight. 

The  perpendicular  distance 
from  the  fulcrum  to  the  direc- 
tions of  the  weight  and  power, 
are  called  the  ar/ra<  of  the  lever. 

362.  An  equilibrium  is  produced  in  all  the  levers,  when  the 
weight  multiplied  by  its  distance  from  the  fulcrum  is  equal  to 
tlie  power  multiplied  bj  its  distance  from  the  fulcrum. 
That  is, 

The  iveight  is  to  the  power,  as  the  distance  froin  the  poxoer  to 
the  fulcrum,  is  to  the  distance  from  the  weight  to  the  fulcrum. 

EXAMPLES. 

1.  In  a  lever  of  the  first  kind,  the  fulcrum  is  placed  at  the 
middle  point :  what  power  will  be  necessary  to  balance  a  weight 
of  40  pounds  ? 

2.  In  a  lever  of  the  second  kind,  the  weight  is  placed  at  the 
middle  point :  what  power  will  be  necessary  to  sustain  a  weight 
of  bOlhs.  ? 

3.  In  a  lever  of  the  third  kind,  the  power  is  placed  at  the 
middle  point :  what  power  will  be  necessary  to  sustain  a  weight 
of  imhs.  ? 

4.  A  lever  of  the  first  kind  is  8  feet  long,  and  a  weight  of 
QOlbs.  is  at  a  distance  of  2  feet  from  the  fulcrum :  what  power 
will  be  necessary  to  balance  it  ? 

362.  When  is  an  equilibrium  produced  in  all  the  levers  !  What  is  then 
the  proportion  between  the  weight  and  power  1 


BG3  MECHANICAL    POWERS. 

5.  In  a  lever  of  the  first  kind,  that  is  6  feet  long,  a  weiglit 
of  200/is.  is  placed  at  1  foot  from  the  fulcrum  :  what  power 
will  balance  it  ? 

6.  In  a  lever  of  the  first  kind,  like  the  common  steelyard, 
the  distance  from  the  weight  to  the  fulcrum  is  one  inch :  at 
what  distance  from  the  fulcrum  must  the  poise  of  lib.  be  placed, 
to  balance  a  weight  of  1Z6.?  A^\eightof^lbs.?  0{2lbs.?Of4:lbs.? 

7.  In  a  lever  of  the  third  kiqd,  the  distance  from  the  fulcrum 
to  the  power  is  5  feet,  and  from  the  fulcrum  to  the  weight 
8  feet :  what  power  is  necessary  to  sustain  a  weight  of  AOlbs.  ? 

8.  In  a  lever  of  the  third  kind,  the  distance  from  the  fulcrum 
to  the  weight  is  12  feet,  and  to  the  power  8  feet :  what  power 
will  be  necessary  to  sustain  a  weight  of  lOOlbs.  ? 

363.  Rejiarks. — In  determining  the  equilibrium  of  the  lever, 
•we  have  not  considered  its  weight.  In  levers  of  the  first  kind, 
the  weight  of  the  lever  generally  adds  to  the  power,  but  in  the 
second  and  third  kinds,  the  weight  goes  to  diminish  the  effect 
of  the  power. 

In  the  previous  examples,  we  have  stated  the  circumstances 
under  which  the  power  will  exactly  sustain  the  weight.  In 
order  that  the  power  may  overcome  the  resistance,  it  must  of 
course  be  somewhat  increased.  The  lever  is  a  very  important 
mechanical  power,  being  much  used,  and  entering,  indeed,  into 
most   other   machines. 

OF   THE    PULLET. 

364.  The  pulley  is  a  wheel,  having  a 
groove  cut  in  its  circumference,  for  the  pur- 
pose of  receiving  a  cord  which  jiasses  over 
it.  When  motion  is  imparted  to  the  cord, 
the  pulley  turns  around  its  axis,  which  is 
generally  supported  by  being  attached  to  a 
beam  above. 

363.  Has  the  wciiiht  been  consiilercd  in  detprmining  the  eoiiilihrijm  of 
the  levers  1  In  a  lever  of  the  first  kind,  will  the  weight  increase  oj  dimii*- 
iih  the  power?     How  will  it  be  in  the  two  other  kiuds  1 

a(i4.  What  is  a  imllcy  ! 


MICCUANIOAL    POWEKS. 


8G9 


365.  Pulleys  are  divided  into  two  kinds,  fixed  pulleys  and 
movable  pulleys.  "When  the  pulley  is  fixed,  it  does  -not  in- 
crease the  power  which  is  applied  to  raise  the  weight,  but 
merely  changes  the  direction  in  which  it  acts. 


3G6.  A  movable  pulley  gives  a  mechan- 
ical   advantage.       Thus,    in    the    movable 
pulley,  the   hand    which  sustains  the  cask 
actually  supports  but  one-half  the  weight 
of  it ;  the  other  half  is  supported  by  the 
hook  to  which  the  other  end  of  the  cord  is 
attached. 


367.  If  we  have  several  movable  pul- 
leys, the  advantage  gained  is  still  greater, 
and  a  very  heavy  weight  may  be  raised  by 
a  small  power.  A  longer  time,  however, 
will  be  required,  than  with  the  single  pulley. 
It  is,  indeed,  a  general  principle  in  machines, 
that  lohat  is  gained  in  2J0wer,  is  lost  in  time  ; 
and  this  is  true  for  all  machines.  There  is 
also  an  actual  loss  of  power,  viz.,  the  resist- 
ance of  the  machine  to  motion,  arising  from 
the  rubbing  of  the  parts  against  each  other, 
which  is  called  the  friction  of  the  machine. 
This  varies  in  the  diiferent  machines,  but 
must  always  be  allowed  for,  in  calculating 
the  power  necessary  to  do  a  given  w^ork.  It 
would  be  wrong,  however,  to  suppose  that 

365.  How  many  kinds  of  pulleys  are  there  ?  Does  a  fixed  pulley  givo 
any  increase  of  power"! 

::!6';.  Does  a  movable  pulley  give  an}'  mechanical  advantage  \  In  a 
eingle  movable  pulley,  how  much  less  is  the  power  than  the  weight? 

'Ml .  Will   an   advantiige  l>s  gained  \iy  several  niovalile  ptilleya  1     Statu 


r 


\ 


VI 


370 


MECUANICAL    POWEKS. 


the  loss  was  equivalent  to  the  gain,  and  that  no  advantage  is 
derived  from  the  mechanical  powers.  We  are  unable  to  aug- 
ment our  strength,  but,  by  the  aid  of  science  we  so  divide  the 
resistance,  that  by  a  continued  exertion  of  power,  we  accom- 
plish that  which  it  would  be  impossible  to  effect  by  a  single 
effort. 

If  in  attaining  this  result,  we  sacrifice  time,  we  cannot  but 
see  that  it  is  most  advantageously  exchanged  for  powei". 

368.  It  is  plain,  that  in  the  movable  pulley,  all  the  parts  of 
the  cord  will  be  equally  stretched,  and  hence,  each  cord  running 
from  pulley  to  pulley,  will  bear  an  equal  part  of  the  weight ; 
consequently. 

The  power  will  always  be  equal  to  the  weight  divided  hy  the 
7iumber  of  cords  which  reach  from  pidley  to  pidley. 

EXAMPLES. 

1.  In  a  single  immovable  pulley,  what  power  will  support  a 
weight  of  QOlbs.  ? 

2.  In  a  single  movable  pulley,  what  power  will  support  a 
weight  of  SOlbs.  ? 

3.  In  two  movable  pulleys  with  4  cords,  (see  last  fig.,)  what 
power  will  support  a  weight  of  lOOlbs.? 

WINCH,    OK    "WHEEL    AND    AXLE. 

369.  This  machine  is  com- 
posed of  a  wheel  or  crank — 
firmly  attached  to  a  cylindri- 
cal axle.  The  axle  is  sup- 
ported at  its  ends  by  two 
pivots,  which  are  of  less 
diameter  than  the  axle  around 
wliich  the  rope  is  coileil,  and 
which  turn  freely  about  the 
points  of  support.  In  order 
to  balance  the  weight,  we  must 
have. 


MECHANICAL    POWERS.  371 

Tlie  power  to  the  weight,  as  the  radius  of  the  axle,  to  the  length 
of  the  crank,  or  radius  of  the  wheel. 

EXAMPLES. 

1.  What  must  be  the  length  of  a  crank  or  radius  of  a  -wheel, 
in  order  that  a  power  of  40/^5.  may  balance  a  weight  of  QOOlbs. 
suspended  from  an  axle  of  6  inches  radius  ? 

2,  "What  must  be  the  diameter  of  an  axle,  that  a  power  of 
lOOlbs.  applied  at  the  circumference  of  a  wheel  of  6  feet  diame- 
ter may  balance  400/fo.  ? 

INCLINED    PLANE. 

380.  The  inclined  plane  is  nothing  more  than  a  slope  or 
declivity,  which  is  used  for  the  purpose  of  raising  weights.  It 
is  not  difficult  to  see  that  a  weight  can  be  forced  up  an  inclined 
plane,  more  easily  than  it  can  be  raised  in  a  vertical  line.  But 
in  this,  as  in  the  other  machines,  the  advantage  is  obtained  by 
a  partial  loss  of  power. 

Thus,  if  a  Aveight  W, 
be  supported  on  the  in- 
clined plane  ABC,  by  a 
cord  passing  over  a  pul- 
ley at  F,  and  the  cord 
from   the    pulley  to    the 

weight  be  parallel  to  the  length  of  the  plane  AB,  the  power  P 
will  balance  the  weight  W,  when 

P  :  W  ::  height  EC  :  length  AB. 

the  general  principle  in  machines.  What  does  the  actual  loss  of  power 
arise  from  1  What  is  this  rubbing  called  1  Does  this  vary  in  different 
machines  I 

36S.  In  the  movable  pulley,  what  proportion  exists  between  the  cord 
and  the  weight  1 

369.  Of  what  is  the  machine  called  the  wheel  and  axle,  composed?  How 
is  the  axle  supported  I  Give  the  proportion  between  the  power  and  tho 
weight. 

370.  What  is  an  inclined  plane  ?  What  proportion  exists  between  the 
power  and  weight  when  they  are  in  equilibrium  1 


372 


MECHANICAL   POWEliS. 


It  is  evident,  that  the  power  ought  to  be  less  than  the  weight, 
since  a  part  of  the  weight  is  supported  by  the  plane :  hence, 

The  power  is  to  the  loeight  as  the  height  of  the  plane  is  to  its 
length. 

exa:^iples. 

1.  The  length  of  a  plane  is  30  feet,  and  its  height  6  feet: 
what  power  will  be  necessary  to  balance  a  weight  of  2Q0lhs.  ? 

2.  The  height  of  a  plane  is  10  feet,  and  the  length  20  feet : 
what  weight  will  a  power  of  bOlhs.  support  ? 

3.  The  height  of  a  plane  is  15  feet,  and  length  45  feet :  what 
power  Avill  sustahi  a  weight  of  l^Olbs.  ? 

THE    WEDGE. 

381.  The  wedge  is  composed  of  two 
inclined  planes,  united  together  along 
their  bases,  and  forming  a  solid  ACB. 
It  is  used  to  cleave  masses  of  wood  or 
stone.  The  resistance  Avhich  it  over- 
comes is  the  attraction  of  cohesion  of 
the  body  which  it  is  employed  to  sepa- 
rate. The  wedge  acts  principally  by  being  struck  with  a  ham- 
mer, or  mallet,  on  its  head,  and  very  little  effect  can  be  produc- 
ed with  it,  by  mere  pressure. 

All  cutting  instruments  are  constructed  on  the  principle  of 
the  inclined  plane  or  wedge.  Such  as  have  but  one  sloi)ing 
edge,  like  the  chisel,  may  be  referred  to  the  inclined  plane,  axid 
such  as  ha\-e  two,  like  the  axe  and  the  knife,  to  the  wedge. 

Half  the  thickness  of  the  head  of  the  wedge  is  to  the  length  of 
one  of  its  sides,  as  the  power  which  acts  against  its  head  to  the 
effect  produced  at  its  side, 

EXAMPLES. 

1.  If  the  head  of  a  wedge  is  4  inches  thick,  and  the  length 


371    \^'liat  is  the  wedge  1     What  is  it  used  for  7     "What  resistance  is  it 
used  to  overcome ! 


MECHANICAL   POWERS.  873 

of  one  of  its  sides   12  inches,  what  will  measure  the  effect  of 
a  force  denoted  by  96  pounds  ? 

2.  If  the  head  of  a  wedge  is  6  inches  thick,  the  length  of 
the  side  27  inches,  and  the  force  applied  measure  by  250  pounds, 
what  will  be  the  measure  of  the  effect  ? 

THE    SCREW. 

381.  The  screw  is  composed 
of  two  parts — the  screw  S,  and 
the  nut  N. 

The  screw  S,  is  a  cylinder 
with  a  spiral  projection  winding 
around  it.  The  nut  N  is  per- 
forated to  admit  the  screw,  and 
within  it  is  a  groove  into  which 
the  thread  of  the  screw  fits 
closely. 

The  handle  D,  which  projects  from  the  nut,  is  a  lever  which 
woj'ks  the  nut  upon  the  screw.  The  power  of  the  screw  depends 
on  the  distance  between  the  threads.  The  closer  the  threads  of 
the  screw,  the  greater  will  be  the  power ;  but  then  the  number 
of  revolutions  made  by  the  handle  D,  will  also  be  proportiona- 
bly  increased ;  so  that  we  return  to  the  general  principle — what 
is  gained  in  power  is  lost  in  time.  The  power  of  the  screw 
may  also  be  increased  by  lengthening  the  lever  attached  to  the 
nut. 

The  screw  is  used  for  compression,  and  to  raise  heavy  w^eights. 
It  is  used  in  cider  and  wine-presses,  in  coining,  and  for  a  variety 
of  other  purposes. 

As  the  distance  between  the  threads  of  a  screw  is  to  the  circum- 
ference of  the  circle  described  hy  the  power,  so  is  the  power  em- 
ployed to  the  weight  raised. 


381.  Of  how  many  parts  is  the  screw  composed  ?  Describe  the  screw. 
What  is  the  thread  \  What  is  the  nut  ?  What  is  the  handle  used  fori  To 
what  uses  is  the  screw  appUed  ''     What  is  the  power  of  tlic  screw  ? 


Oli  QUESTIONS    IN    PHILOSOPHT. 

EXAMPLES. 

1.  If  the  distance  between  the  thi-eads  of  a  screw  is  half  an 
inch,  and  the  circumference  described  by  the  handle  15  feet, 
what  weight  can  be  raised  bj  a  power  denoted  by  720  pounds  ? 

2.  If  the  threads  of  a  screw  are  one-third  of  an  inch  apart, 
and  the  handle  is  12  feet  long,  what  power  must  be  applied  to 
sustain  2  tons  ? 

3.  "What  force  applied  to  the  handle  of  a  screw  10  feet  long, 
>vith  threads  1  inch  apart,  working  on  a  wedge  whose  head  is 
5  inches,  and  length  of  side  30  inches,  will  produce  an  effect 
measured  by  lOOOOlbs.  ? 

4.  If  a^ower  of  300  pounds  applied  at  the  end  of  a  lever 
15  feet  long  will  sustain  a  weight  of  282744/^5.,  what  is  the 
distance  between  the  threads  of  the  screw  ? 


QUESTIONS   IN   NATURAL   PHILOSOPHY. 
UNIFORM    MOTION. 

382.  If  a  moving  body  passes  over  equal  spaces  in  equal 
successive  small  portions  of  time,  it  is  said  to  move  with  uni- 
form motion,  or  uniformly. 

383.  The  velocity  of  a  moving  body  is  measured  by  the 
space  passed  over  in  a  second  of  time. 

384.  The  space  passed  over  in  any  time  is  equal  to  the  pro- 
duct of  the  velocity  multiplied  by  the  number  of  seconds  in  the 
time. 

If  we  denote  the  velocity  by  V,  the  space  passed  over  by  S, 
and  the  time  by  T,  we  have 

S  =  V  X  T. 


382.  What  is  a  unilorm  motion  1 

383.  Wliat  is  the  velocity  of  a  moving  body  1 

384.  To  wliat  is  tl)C  space  passed  over  in  a  unit  of  time  equal  1     What 
lb  the  span;  passed  over  equal  to,  in  uniform  motion  ' 


QUESTIONS   IN    PHILOSOPHY.  375 

EXAMPLES. 

1.  A  steamboat  moves  with  a  velocity  of  23  feet:  what  space 
does  it  pass  over  in  1\  hours  ? 

2.  A  locomotive  is  moving  with  a  velocity  of  32  feet :  what 
distance  will  it  travel  in  3  minutes  ? 

3.  A  horse  travels  uniformly  a  distance  of  12  miles  with  a 
velocity  of  6  feet :  what  time  does  he  require  to  perform  the 
journey  ? 

4.  A  carriage  performs  a  journey  of  15  miles  in  2|-  hours : 
with  what  velocity  does  it  move  ? 

5.  The  hammer  of  a  pile-driver  is  moved  upwai'd  a  distance 
of  *35  feet  with  a  velocity  of  1-^  feet :  what  time  is  required  to 
raise  it  ? 

6.  A  ton  of  coal  is  raided  from  a  mine  1000  feet  deep  in  3^ 
minutes  :  with  what  velocity  does  it  move  ? 

7.  A  vessel  containing  a  criminal,  after  leaving  a  port,  sailed 
with  a  daily  speed  of  170  miles  ;  four  days  after,  a  clipper  was 
dispatched  in  pursuit,  and  sailed  at  a  daily  rate  of  275  miles  : 
in  what  time  did  the  clipper  overtake  the  vessel  ? 

8.  A  bird  flew  a  distance  of  100  miles  in  11  hours:  with 
what  velocity  did  it  travel  ? 

9.  Sound  moves  with  a  velocity  of  1127  feet.  If  the  report 
of  a  gun  was  heard  31.3  seconds  after  the  flash  was  seen,  what 
distance  was  the  gun  from  the  observer  ? 

10.  A  hurricane  moves  with  a  velocity  of  95  feet :  what  time 
does  it  take  to  move  through  3  degrees  of  latitude,  the  degree 
being  estimated  at  G91  miles  ? 

11.  The  velocity  of  light  has  been  found,  by  astronomical 
observations,  and  by  experiments  made  in  France,  to  be  191,300 
miles  :  what  time  will  it  occupy  to  traverse  the  mean  distance 
of  the  earth  from  the  sun,  or  95000000  of  miles  ? 

12.  If  a  message  sent  by  electro-magnetic  telegraph  2300 
miles  requires  14  seconds  for  its  transmission,  what  is  the 
velocity  of  the  magnetic  current  in  this  telegraph  line  .'' 


376  QUESTIONS    IN    PHILOSOPHY. 

LATVS    OF    FALLING   BODIES. 

385.  A  body  fulling  vertically  downward  iu  a  vacuum,  falls 
tlirough  16^2^;'.  during  the  first  second  after  leaving  its  place 
of  rest,  481//.  during  the  second  second,  SOj^^ft.  the  third 
second,  and  so  on  :  the  spaces  forming  an  arithmetical  pro- 
gression of  which  the  common  difference  is  32^fL,  or  double 
the  space  fallen  through  during  the  first  second.  This  number 
is  called  the  measure  of  the  force  of  gravity,  and  is  denoted 

o<S6.  It  is  seen  from  the  above  that  the  velocity  of  a  body  is 
continually  increasing.  If  H  denote  the  height  fallen  through, 
T,  the  time,  V,  the  velocity  acquired,  and  c/,  the  force  of  gravity, 
the  following  formulas  have  been  found  to  express  the  relations 
between  these  quantities  : 

V    =    ^    X  T  ...  (1). 

V2  =  2r/    X  H  .     .     .  (2). 

H  =  IV  X  T  ...  (3). 

B.  =ig    xT^  .     .     .  (4). 

From  which  we  see, 

1st.  That  the  velocity  acquired  at  the  end  of  any  time,  is  equal 
to  the  force  of  gravity  (32|^)  multiplied  by  the  time. 

2d.  Tliat  the  square  of  the  velocity  is  equal  to  twice  the  force 
of  gravity  multiplied  by  the  height;  or,  the  velocity  is  equal  to 
the  square  root  of  that  quantity, 

3d.  2'hat  the  space  fallen  through  is  equal  to  one-half  the 
velocity  multiplied  by  the  time. 

4th.  2Viat  the  sjmce  fallen  through  is  equal  to  one-half  the  force 
of  gravity  multij^lied  by  the  square  of  the  time. 

385.  If  a  body  falls  vertically,  in  a  vacuum,  how  far  will  it  fall  in  tlio 
first  serond  of  time  ?  How  far  on  the  second  second  second  !  In  the 
third  ]  What  is  the  common  ditlercnce  of  the  spaces  1  M'hat  is  the 
measure  of  the  force  of  gravity? 

386.  How  does  the  velocity  of  a  falling  body  change  ?  What  is  the 
velocity  acquired  at  the  end  of  any  time  equal  to  ?  What  is  the  space 
(alien  through  equal  to  ? 


QUESTIONS    IN    PHILOSOPHY.  377 

387.  If  a  body  is  thrown  vertically  upwards  in  a  vacuum,  its 
motion  will  be  continually  retaixled  by  the  action  of  gravitation. 
It  will  finally  reach  the  highest  point  of  its  ascent,  and  then 
begin  to  descend.  The  height  to  which  it  will  rise  may  be 
found  by  the  second  formula  in  the  preceding  paragraph,  when 
the  velocity  with  which  it  is  projected  upward  is  known  ;  for 
the  times  of  ascent  and  descent  will  be  equal. 

38S.  The  above  laws  are  only  approximately  true  for  bodies 
falling  through  the  air,  in  consequence  to  its  resistance.  We 
may  measure  the  depths  of  wells  or  mines  and  the  heights  of 
elevated  objects  approximately  by  using  dense  bodies,  as  leaden 
bullets  or  stones,  which  present  small  surface  to  the  air. 

EXAMPLES. 

1.  A  body  has  been  falling  12  seconds  :  what  space  has  it 
described  in  the  last  second,  and  what  in  the  whole  time  ? 

2.  A  body  has  been  falling  15  seconds :  find  the  space 
described  and  the  velocity  acquired. 

3.  How  far  must  a  body  fall  to  acquire  a  velocity  of  120 
feet? 

4.  How  many  seconds  wiU  it  take  a  body  to  fall  through  a 
space  of  100  feet  ? 

5.  Find  the  space  through  which  a  heavy  body  falls  in  10 
seconds,  and  the  velocity  acquired. 

G.  How  far  must  a  body  fall  to  acquire  a  velocity  of  1000 
feet? 

7.  A  stone  is  dropped  into  a  well  and  strikes  the  M'ater  in 
32  seconds  :  what  is  the  depth  of  the  well  ? 

8.  A  stone  is  di-opped  from  the  top  of  a  bridge  and  strikes 
the  water  in  2.5  seconds  :  what  is  the  heio-ht  of  the  bridge  ? 

9.  A  body  is  thrown  vertically  upward  with  a  A'elocity  of 
1  GO  feet :  what  height  will  it  reach,  and  what  wull  be  the  time 
of  ascent  ? 

387.  How  far  will  a  body  ascend  when  projected  upwards  1 

388.  Arc  the  above  laws  perfectly  or  only  approximately  true ' 

17 


378  (iUESTIOXS    IN   PHILOSOPHY. 

10.  An  arrow  shot  perpendicularly  upwards  returned  again 
In  10  seconds.  Required  the  velocity  with  which  it  was  shot, 
and  the  height  to  which  it  rose. 

11.  If  a  body  falls  freely  in  vacuum,  what  will  be  its  velocity 
after  45  seconds  of  fall  ? 

12.  During  how  many  seconds  must  a  body  fall  in  a  vacuum 
to  acquire  a  velocity  of  1970  feet,  which  is  that  of  a  cannon  ball? 

13.  What  time  is  required  for  a  body  to  fall  in  a  vacuum, 
from  an  elevation  of  3280  feet? 

14.  From  what  height  must  a  body  fall  to  acquire  a  velocity 
of  984  feet  ? 

15.  A  rocket  is  projected  vertically  upward  with  a  velocity 
of  386  feet :  after  what  time  will  it  begin  to  fall,  and  to  what 
height  will  it  rise  ? 

SPECIFIC    GKAVITT. 

389.  The  SPECIFIC  gravity  of  a  body  is  the  weight  of  a 
unit  of  volume  in  terms  of  a  unit  of  the  standard.  Distilled 
rain  water  is  the  standard  for  measuring  the  specific  gravity  of 
bodies.  Thus,  1  cubic  foot  of  distilled  rain  water  weighs  1000 
ounces  avoirdupois.  If  a  piece  of  stone,  of  the  same  volume, 
weighs  2500  ounces,  its  specific  gravity  is  2.5 ;  that  is,  the 
stone  is  2.5  times  as  heavy  as  water. 

If,  then,  we  denote  the  standard  by  1,  the  specific  gravity  of 
all  other  bodies  will  be  expressed  in  terms  of  this  standard ; 
and  if  we  multiply  the  number  denoting  the  specific  gravity  of 
any  body  by  1000,  the  product  will  be  the  weight  in  ounces  of 
1  cubic  foot  of  that  bod}'. 

If  any  body  be  Aveighed  in  air  and  then  in  water,  it  will 
weigh  less  in  water  than  in  air.  The  difl:erence  of  the  weights 
will  be  equal  to  the  sustaining  force  of  the  watei",  which  is  found 
to  be  equal  to  the  weight  of  an  equal  volume  of  water  :  hence, 

389.  What  is  the  specific  gravity  of  a  body'!  What  is  the  standard  for 
mpasurin<^  the  specific  gravity  of  a  body  ■  Wliat  is  the  minirrical  value 
of  the  cubic  foot  of  a  body  !  How  do  you  find  the  specific  gravity  of  a 
body  ? 


QUESTIONS   IN   PHILOSOPHr. 


379 


If  we  knoio  or  can  find  the  iveight  of  a  body  in  air  and  in 
water,  the  difference  of  these  tveights  will  he  equal  to  that  of  an 
equal  volume  of  water  ;  and  the  weight  of  the  body  in  air  divided 
by  this  difference  will  be  the  measure  of  the  specific  gravity  of 
the  body,  compared  with  water  as  a  standard. 


TABLE 

OP    SPECIFIC    GRAVITIES. WATER    1. 


NAMES    OP    BODIES. 


SPEC.   GRAV. 


NAMES    OF    BODIES.         SPEC.   GRAV 


METALS 
Platinum,  . 
Gold,  .     .     . 
Quicksilver, 
Lead,  . 
Silver,   . 
Copper,    . 
Bronze, 
Brass,  . 
Steel, 

L'on,    .     .     . 
Tin, 
Zinc,   .     .     . 


BUILDING    STONES 

Hornblende,    .     . 
Basalt, 

Alabaster,       .     . 
Syenite,    .     .     . 
Dolerite,     .     .     . 
Gniess, 

Quartz,       .     .     . 
Limestone,    . 
Phonolite,  . 
Granite,    . 
Stone  for  building, 
Trachytse, 


21.000 

19.500 

13.500 

11.350 

10.51 

8.800 

8.758 

8.000 

7.800 

7.500 

7.291 

7.215 


3.10 
3.10 
3.00 
3.00 
2.93 
2.90 
2.75 
2.72 
2.69 
2.66 
2.62 
2. GO 


Porphyry,    . 
Sandstone, 
Brick,     .     .     . 

"WOODS. 

Oak,  fresh  felled. 
White  Willow, 
Box, 
Elm, 

Hanbeam, 
Larch,    . 
Pine,    . 
Maple,   . 
Ash,     .     . 
Birch, 
Fir,  .     .     . 
Horse  Chestnut, 

SOLID  BODIES 

Common  earth, 
Moist  sand. 
Clay,    .     .     , 
Flint,     .     .     . 
Ice,  .... 
Lime,     .     .     . 
Tallow,     .     . 
Wax,      .     .     . 


2.60 
2.50 
1.86 


1.049 

0.9859 
0.9822 
0.9476 
0.9452 
0.9206 
0.9121 
0.9036 
0.9036 
0.9012 
0.8941 
0.8614  ,' 


1.480 
2.050 
2.150 
2.542 
0.926 
1.842 
0.942 
0.969 


By  inspecting  this  Table,  we  see  the  Aveight  of  each  body 
compared  with  an  equal  volume  of  water.  Thus,  platina  is 
21  times  as  heavy  as  water;  gold,  19  times  as  heavy;  iron,  7^ 
times  as  heavy,  &c. 


3 so  QUESTIONS    IN    PHILOSOPHY. 

EXAMPI-ES    ILLUSTRATING    SPECIFIC    GRAVITT. 

1.  A  piece  of  copper  weighs  93  grains  in  air,  and  82^  grains 
in  water:  what  is  its  specific  gravity? 

2.  How  many  cubic  feet  are  there  in  2240  pounds  of  dry  oak, 
of  which  the  specific  gravity  is  .925,  a  cubic  foot  of  standard 
•water  Aveiajhing  1000  ounces  ? 

3.  A  piece  of  pumice  stone  weighs  in  air  50  ounces,  and 
w'hen  it  is  connected  with  a  piece  of  copper  whicli  weighs  390 
ounces  in  air,  and  345  ounces  in  water,  the  compound  weighs 
344  ounces  in  water :  what  is  the  specific  gravity  of  the  stone  ? 

4.  A  right  prism  of  ice,  the  length  of  whose  base  is  20.45 
yards,  breadth  15.75  yards,  and  height  10.5  yards,  floats  on 
the  sea;  the  specific  gravity  of  the  ice  is  .930,  and  that  of  the 
sea  water  1.026  :  Avhat  is  the  height  of  the  prism  above  the  sui'- 
face  of  the  water  ? 

5.  A  vessel  in  a  dock  was  found  to  displace  6043  cubic  feet 
of  water :  what  was  the  weight  of  the  vessel,  each  cubic  foot 
of  the  water  weighing  G3  pounds  ? 

6.  A  piece  of  glass  was  found  to  weigh  in  the  air  33  ounces, 
and  in  the  water  21  ounces  :  wliat  was  its  specific  gravity  ? 

7.  A  piece  of  zinc  weighed  in  the  air  17  pounds,  and  lost 
when  weighed  in  water  2.35  pounds :  what  was  its  specific 
gravity  ? 

8.  If  a  piece  of  glass  weighed  in  water  loses  318  ounces  of 
its  weight,  and  weighed  in  alcoliol  loses  250  ounces,  Avhat  is  the 
specific  gravity  of  the  alcohol  ? 

9.  A  flask  filled  with  distilled  water  weighed  14  ounces  ; 
filled  with  brandy,  it  weighed  13.25  ounces ;  the  flask  itself 
weighed  8  ounces  :  what  was  the  specific  gravity  of  the  brandy? 

10.  What  is  the  weight  of  a  cubic  foot  of  statuary  marble, 
of  which  the  specific  gravity  is  2.837,  the  cubic  foot  of  water 
weighing  1000  ounces? 

11.  A  jar  containing  air  weighed  24  ounces  33  grains;  tha 
air  was  tlien  excluded,  and  the  jar  weighed  24  ounces  ;  the  jar 
being  tlieu  filled  with  yxygen  gas  weighed   24  ounces    3G.4 


QUESTIONS    IN    PHILOSOPHY.  881 

grains :  wliat  was   the  specific  gravity  of  the  oxygen,  the  air 
being  taken  as  the  standard  ? 

12.  A  cylindrical  vase  having  a  base  whose  interior  diameter 
is  4  inches,  stands  upon  a  horizontal  plane  :  2G.'2  pounds  of 
mercury  is  poured  into  the  vase.  Kequired  the  height  to  wliich 
the  liquid  will  rise,  the  specific  gravity  of  mercury  being  13.59G. 

13.  A  piece  of  alabaster  weighs  in  the  air  7.55  grains,  in  the 
water  5.17  grains,  and  in  another  liquid  G.35  grains  :  what  is 
the  specific  gravity  of  the  alabaster  and  of  the  liquid  ? 

14.  What  effort  will  be  required  to  prevent  a  cubic  inch  of 
platinum,  immersed  in  mercury,  from  sinking,  the  specific 
gravity  of  the  platinum  being  21,5,  and  that  of  the  mercury 
13.G  ? 

15.  What  weight  of  mercury  will  a  conical  vase  contain  of 
which  the  radius  of  the  base  is  9  inches  and  the  altitude  34 
inches,  the  specific  gravity  of  the  mercury  being  13.59G  ? 

mariotte's  law. 

390.  This  laAV,  which  relates  to  air  and  all  other  gases,  steam, 
and  all  other  vapors,  was  discovered  by  the  abbe  Mariotte,  a 
French  philosopher,  who  died  in  1G84.  It  will  be  easily  under- 
stood from  a  particular  example. 

Suppose  an  upright  cylindrical  vessel  in  a  vacuum  contains  a 
gas  which  is  confined  in  the  vessel  by  a  piston  at  the  upper  end. 
Suppose  tlie  gas  or  vapor  fills  the  whole  vessel,  and  the  piston  is 
loaded  with  a  weight  of  5  pounds.  If  now,  the  piston  be  loaded 
with  a  weight  of  10  pounds,  the  gas  will  be  compressed  and 
occupy  only  half  its  former  space.  If  the  weight  be  increased 
to  15  pounds,  the  gas  will  have  only  one-third  of  its  original 
volume,  and  so  on.  At  the  same  time,  the  density  of  the  gas 
or  vapor  will  be  doubled,  made  three  times  as  great,  and  so  on. 
The  law,  therefore,  may  be  thus  stated : 


.190.  To  what  is  the  vokime  uf  a  vapor  or  gas   proportional  1     To  what 
is  its  density  proportional  1 


382  QUESTIONS    IN    PHILOSOPHY. 

The  temjyeratwe  remaining  the  same,  the  volume  of  a  gas  or 
va2wr  is  inversehj  proportional  to  the  2yrcssure  which  it  sustains. 
Also,  the  density  of  a  gas  or  vapor  is  directly  p>yop)ortional  to 
the  pjresmre. 

EXAMPLES. 

1.  A  vase  contains  4.3  quarts  of  air,  the  pressure  being  10 
pounds  :  what  will  be  the  volume  of  the  air  when  the  pressure 
is  12.3  pounds,  the  temperature  remaining  the  same? 

2.  Under  a  pressure  of  15  pounds  to  the  square  inch,  a  cer- 
tain quantity  of  gas  occupies  a  volume  of  20  quarts :  what 
pressure  must  be  applied  to  reduce  the  volume  to  8  quarts  ? 

3.  A  quart  of  air  weighs  2.6  grains  under  a  pressure  of  15 
pounds  :  what  will  be  the  weight  of  a  quart  if  the  pressure  be 
reduced  to  14.2  pounds  ? 

4.  The  jjressure  upon  the  steam  contained  in  a  cylinder  is 
increased  from  25  pounds  upon  the  square  inch  to  47  pounds : 
what  part  of  the  original  volume  will  be  occupied  ? 

5.  How  will  the  density  of  the  steam  in  the  last  example,  at 
the  second  pressure,  compare  with  that  at  the  first  ? 

6.  Eight  quarts  of  hydrogen  gas  are  contained  in  a  vessel  and 
submitted  to  a  pressure  of  22  pounds  :  how  many  quarts  of  ga? 
will  there  be  if  the  pressure  is  changed  9i  pounds .'' 


APPENDIX. 


DIFFERENT    KINDS    OF    UNITS. 

391.  There  are  eight  kinds  of  units  : 
1st.  Abstract  Units  ; 

2d.    Units  of  Currency  or  Coin  ; 

3d.    Linear  Units,  or  Units  of  Length  ; 

4th.  Units  of  Surface,  or  Superficial  Units ; 

5th.  Units  of  Volume,  including  Cubic  Units  and  Gallons ; 

Gth.  Units  of  "Weight ; 

7th.  Units  of  Time  ;  and 

8th.  Units  of  Circular  or  Angular  Measure. 

ABSTRACT    UNITS. 

392.  The  abstract  unit  1  is  the  base  of  all  numbers,  and  is 
called  a  unit  of  the  first  order.  The  unit  1  ten  is  a  unit  of  the 
second  order  ;  the  unit  1  hundred  is  a  unit  of  the  third  order ; 
and  so  for  units  of  the  higher  orders.  These  are  abstract  num- 
bers formed  from  the  unit  1,  according  to  the  scale  of  tens.  All 
abstract  integral  numbers  are  collections  of  these  units. 

UNITS    OF    CURRENCY. 

393.  In  all  civilized  and  commercial  countries,  great  care  is 
taken  to  fix  a  standard  value  for  money,  which  standard  is 
called  the  Unit  of  Currency. 

In  the  United  States,  the  unit  of  currency  is  1  dollar  ;  in 
Great  Britain  it  is  1  pound  sterling,  equal  to  S4,84  ;  in  France 
it  is  1  franc,  equal  to  18j  cents.  All  sums  of  money  are 
expressed  in  the  unit  of  currency  or  in  units  derived  from  the 
unit  of  currency,  and  having  fixed  proportions  to  it. 

391.   How  many  kinds  of  units  are  there  in  Arithmetic  1     Name  them. 

393.  What  is  said  of  the  abstract  unit  1  1  AVhat  is  a  unit  of  the  2J 
order  I  What  of  the  3d  \  4th  1  5th  !  &c.  How  are  these  numbers 
("ormed  from  1  ! 


884:  APPEjroiX. 

UNITS  OF  LENGTH. 

394.  One  of  the  most  important  units  of  measure  is  that  fof 
distances,  or  for  the  measurement  of  length.  A  practical  want 
has  ever  been  felt  of  some  fixed  and  invariable  standard  with 
which  all  distances  may  be  compared  :  such  fixed  standard  ha3 
been  sought  for  in  nature. 

There  are  two  natural  standards,  either  of  which  affords  this 
desired  natural  element.  Upon  one  of  them,  the  English  have 
founded  their  system  of  measures,  from  which  ours  is  taken, 
and  upon  the  other,  the  French  have  based  their  system.  Tliese 
two  systems,  being  the  only  ones  of  importance,  will  be  alone 
considered. 

395.  First. — The  English  system  of  measures,  to  which 
ours  conforms,  is  based  upon  the  law  of  nature,  that  the 
force  of  gravitij  is  constant  at  the  same  jJoint  of  the  carlh\s  sur- 
face, and  consequently,  that  the  length  of  a  pendulum  which 
oscillates  a  certain  number  of  times,  in  a  given  period,  is  also 
constant.  Had  this  unit  been  known  brfore  the  adoption  and 
use  of  a  system  of  measures,  it  would  liave  formed  the  natural 
unit  for  division,  and  been  the  natural  base  of  the  system  of 
linear  measure.  But  the  foot  and  inch  had  long  been  used 
as  units  of  linear  measure  ;  and  hence,  the  length  of  tlie  pen- 
dulum, the  new  and  invariable  standard,  was  expressed  in  terms 
of  the  known  units,  and  found  to  be  equal  to  39.1393  inches. 
The  new  unit  was  therefore  declared  invariable — to  contain 
39.1393  equal  parts,  each  of  which  was  called  an  inch  ;  12  of 
these  parts  were  declared  by  act  of  Parliament  to  be  a  standard 
foot,  and  36  of  them,  an  Imperial  yard.  The  Imperial  yard 
and  the  standard  foot  are  marked  ujwn  a  brass  bar,  at  tlie  tem- 
perature of  62^-°,  and  these  are  the  linear  measures  from  which 


393.  "What  is  a  unit  of  ciiiTcncy  1     "What  is  the  unit  of  currency  in  tho 
Unitcii  St;itos?     Wliat  in  (iroat  Britain  !     "What  in  Fiance  1 

394.  For  what  is  an  invaiialile  stanilanl  of  lenglli  used  ! 

395.  What  is  the  standard  unit  of  length  in  the  EngUsh  system  1    What 
in  ours  ■> 


UNITS   OF   SURFAOK.  8  85 

ours  are  taken.  The  comparison  has  been  made  by  means  of 
a  brass  scale  82  inches  long,  manufactured  by  Troughton  in 
London,  and  now  in  the  possession  of  the  Treasury  Depart- 
ment. 

396.  Second. — The  French  system  of  measures  is  founded 
upon  the  principle  of  the  invariability  of  the  length  of  an  arc 
of  the  same  meridian  between  two  fixed  points.  By  a  very 
minute  survey  of  the  length  of  an  arc  of  the  meridian  from 
Dunkirk  to  Barcelona,  the  length  of  a  quadrant  of  the  meri- 
dian was  computed,  and  it  has  been  decreed  by  the  French  law 
that  the  ten-millionth  part  of  this  length  shall  be  regarded  as 
a  standard  French  metre,  and  from  this,  by  multiplication  and 
division,  the  entire  system  of  linear  measures  has  been  estab 
Ilshed. 

On  comparing  two  scales,  vei'y  accurately,  it  has  been  found  that 
the  French  metre  is  equal  to  39.37079  English  inches — differing 
somewhat  from  the  English  yard.  This  relation  enables  us  to 
convert  all  measures  in  either  system  into  the  corresponding 
measures  of  the  other. 

UNITS   OF   SURFACE. 

397.  The  linear  unit  having  been  established,  the  most  con- 
venient UNIT  OF  SURFACE  is  the  area  of  a  square,  one  of  whose 
sides  is  the  unit  of  length.  Thus,  the  units  of  surface  in  com- 
mon use,  are 

A  square  inch  =  a  square  on  1  inch. 
A  square  foot   =144  square  inches. 
A  square  yard  =  9  square  feet. 
A  square  rod    =  30^  square  yards. 
&c.  &c. 

396.  What  is  the  standard  unit  of  length  in  the  French  system  1  IIow 
was  it  found  1  How  docs  the  French  metre  compare  with  the  Imperial 
yard  1 

397.  What  is  the  most  convenient  unit  of  surfaced  What  are  those  in 
common  use  I 


386  Al'PENDIX. 

UNITS   OF   VOLUME. 

398.  The  Unit  of  Volume,  for  the  measurement  of  solids,  h 
taken  equal  to  the  volume  of  a  cube  one  of  whose  edges  is  equal 
to  the  linear  unit.     The  units  of  volume  in  common  use  are 

A  cubic  inch  =  a  cube  whose  edge  is  1  inch ; 
A  cubic  foot  =  a  cube  whose  e  ige  is  1  foot  =  1728  cubic  in, 
A  cubic  yard=:  a  cube  whose  edge  is  1  yard  =  27  cubic  feet. 
A  perch  of  stone  =  24f  cubic  feet ; 

or  a  block  of  stone  1  rood  long,  1  foot  thick,  and  lA  feet  wide. 

The  standai-d  unit  of  volume  for  the  measurement  of  liquids 
i3  the  wine  gallon,  which  contains  231  cubic  inches. 

The  standard  i( nit  of  dry  measure  is  the  Winchester  bushel, 
which  contains  2150.4  cubic  inches,  nearly. 

UNITS   OF   WEIGHT. 

399.  Having  fixed  an  invariable  unit  of  length,  we  passed 
easily  to  an  invariable  unit  of  surface,  and  then,  to  an  invariable 
unit  of  volume.  We  wish  now  to  define  an  invariable  unit  of 
weight. 

It  has  been  found  that  distilled  rain  water  is  the  most  inva- 
riable substance  ;  hence,  this,  at  a  given  temperature,  has  been 
adopted  as  the  standard. 

We  have  two  units  of  weight,  the  avoirdupois  pound,  and  the 
pound  troy. 

The  standard  avoirdupois  pound  is  the  weight  of  27.701554. 
cubic  inches  of  distilled  water. 

The  standard  Troy  -pound  is  the  weight  of  22.794422  cubic 


398.  What  is  thp  unit  of  volume  for  the  measurement  of  solids  \  AVhat 
are  tiiose  in  common  use  \  ^\'h,^t  is  the  standard  unit  for  the  iiic;i.suro« 
ment  of  liquids  !     What  for  dry  measure  ! 

399.  What  is  used  as  a  standard  in  fixing  the  iinits  of  weight?  How 
many  units  of  weight  have  we  1  How  is  the  .standard  avoirdupois  iKumd 
determined  ?  How  tlie  Troy  pound  !  Wliich  is  represented  by  a  standard 
at  the  mint  1 


UNITS    OF   TIME.  387 

inches  of  distilled  rain  water.  This  standard  is  at  present  kept 
in  the  United  States  Mint  at  Philadelphia,  and  is  the  standard 
unit  of  weight. 

UNITS   OF   TIME. 

400.  Time  can  only  be  measured  by  motion.  The  diurnal 
revolution  of  the  earth  affords  the  only  invariable  motion  ; 
hence,  the  time  in  which  it  revolves  once  on  its  axis,  is  the 
natural  unit,  aud  is  called  a  day.  From  the  day,  by  multi- 
plication, we  form  the  weeks,  mouths  and  years  ;  and  by  divi- 
sion, the  hours,  minutes  and  seconds. 


UNITS    OF   CIRCULAR   OR  ANGULAR   MEASURE. 


401.  This  measure  is  used  for  the  measurement  of  an<Tles, 
and  the  natural  unit  is  the  right  angle.  But  this  is  not  the 
most  convenient  unit.  The  unit  chiefly  used  is  the  360  part 
of  the  circumference  of  a  circle,  called  a  degree,  which  is  divided 
into  60  equal  parts  called  minutes,  and  these  again  into  60  equal 
parts  called  seconds. 

REJIARKS. 

402.  It  is  seen  that  all  the  units,  determined  by  the  pendu- 
lum, depend  on  time  as  the  ultimate  base  ;  that  is,  the  length 
of  a  pendulum  which  A\'ill  vibrate  seconds  determines  all  the 
units  of  measure  and  weight. 

Now,  time  is  measured  by  motion,  and  the  motion  of  the 
earth  on  its  axis  is  the  only  invariable  m,otion.  Hence,  we  refer 
to  this  to  fix  the  unit  of  time,  on  which  the  unit  of  length 
depends,  and  from  which  all  the  other  units  are  derived. 

403.  No  class  of  pupils  can  rightly  and  clearly  apprehend 
the  nature  of  numbers  and  the  operations  performed  upon  them, 

400.  How  is  time  measured  ?  "What  motion  is  uniform  1  ^^'hat  is  tho 
natural  unit  ? 

401    For  what  is  circular  or  angular  measure  used  1     What  is  the  uniti 

402.  On  what  do  the  units  determined  by  the  pendulum  depend  !  How 
i    time  measured  ]     To  what  then  arc  all  these  units  referred  1 

403.  How  are  the  ideas  of  the  absolute  and  relative  values  of  the  units 
to  be  communicated  to  a  class  1     What  apparatus  is  necessary  ^ 


388  APPENDIX. 

without  distinct  and  fixed  notions  of  the  units  ;  hence,  every 
teacher  should  labor  to  point  out  their  absolute  and  relative 
values  :  this  can  only  be  done  b}''  means  of  sensible  objects. 

Every  school  room,  therefore,  should  be  provided  with  a 
complete  set  of  all  the  denominate  units.  The  inch,  the  foot, 
the  yard,  the  rod,  should  be  accurately  marked  off  on  a  con- 
spicuous part  of  the  room,  together  with  the  principal  units  of 
sui'face,  the  square  inch,  square  foot,  square  yard,  &c. 

The  units  of  volume  should  also  be  exhibited.  The  cubic 
inch  and  the  cubic  foot  will  serve  as  illustrations  for  one  class 
of  the  units  of  volume  ;  and  the  j^int,  quart,  gallon  and  bushel, 
should  be  exhibited  to  illustrate  the  others. 

The  unit  of  weisrht  should  also  be  seen  and  handled.  A 
child  even  can  apprehend  what  is  meant  by  an  ounce  or  a  2^oiind 
when  it  takes  one  of  these  weights  in  its  hand ;  and  mature 
years  can  acquire  the  idea  in  no  other  way. 

Let,  therefore,  every  school  room  be  furnished  Avith  a  com- 
plete set  of  models  to  illusti-ace  and  explain  the  absolute  and 
relative  values  of  the  different  units. 

UNITED    STATES    MONEY. 

404.  United  States  Monet  is  the  currency  established 
by  Congress,  A.  D.  1786.  The  names  or  denominations  of  its 
units  are.  Eagles,  Dollars,  Dimes,  Cents,  and  Mills. 

The  coins  of  the  United  States  are  of  gold,  silver,  and  cop- 
per, and  are  of  the  following  denominations  : 

1.  Gold  :  Eagle,  half-eagle,  three-dollars,  quarter-eagle,  dollar. 

2.  Silver :  Dollar,  half-dollar,  quarter-dollar,  dime,  half-dime, 
and  three-cent  piece. 

3.  Copper  :  Cent,  half  cent. 

TABLE. 

10  IMills     make  1  Cent,  marked  ct. 

10  Cents     -     -  1  Dime,     -     -    d. 

10  Dimes    -     -  1  l^Jollar,  -     -    $. 

lU  Dollars  -     -  1  Eagle,    -     -    B. 


UNITED    STATES    MONET.  889 


Mills. 

Cents. 

Dimes. 

Dollars. 

Eagles. 

10 

=  1 

100 

=  10 

=  1 

1000 

=  100 

z=  10 

=  1 

(0000 

=  1000  ^ 

=  100 

=  10 

=  1 

105.  It  is  seen,  fix)m  the  above  table,  that  in  United  States 
tuviiey,  the  2^r»?2ary  unit  is  1  mill ;  that  the  units  of  the  scale, 
in  passing  from  mills  to  cents,  are  10.  The  second  unit  is  1  cent, 
and  the  units  of  the  scale,  in  passing  to  dimes,  are  10.  The  third 
unit  is  1  dime,  and  the  units  of  the  scale,  in  passing  to  dollars, 
are  10.  The  fourth  unit  is  1  dollax%  and  the  units  of  the  scale,  in 
passing  to  eagles,  are  10.  This  scale  is  the  same  as  in  siiiq^le 
numbers;  therefore, 

27^6  nnits  of  United  States  money  may  he  added,  suhtracted, 
rutdtipUed,  and  divided  hy  the  same  rides  as  are  appUcahle  to 
simple  numbers. 

Notes. — The  present  standard  or  degree  of  purity  of  the  coins 
was  fixed  by  Act  of  Congress  in  1837.     It  is  this  : 

1.  Nine  hundred  equal  parts  of  pure  gold,  are  mixed  with  100  parts 
of  alloy,  of  copper  and  silver,  (of  which  not  more  than  one-half 
must  be  silver)  thus  forming  1000  parts,  equal  to  each  other  in 
weight.  The  silver  coins  contain  900  parts  of  pure  silver,  and  100 
parts  of  pure  copper.     The  copper  coins  are  of  pure  copper. 

2.  The  eagle  contains  258  grains  of  standard  gold,  and  the  other 
gold  coins  in  the  same  proportion.  The  dollar  contains  412^  grains 
of  standard  silver,  and  the  others  in  the  same  proportion  The  cent, 
168  grains  of  puro  copper. 

3.  If  a  given  quantity  of  gold  or  silver  be  divided  into  24  equal 
parts,  each  part  is  called  a  ca;rat.  If  any  number  of  carats  be  mixed 
with  so  many  equal  caratt,  nf  a  less  valuable  metal,  that  there  be  24 


404.  What  is  United  Slat  ;s  money  1  What  are  the  names  of  its  units  1 
What  are  the  coins  of  the  Lnited  States  1  Which  gold  1  Which  silver  1 
Which  copper  1 

405.  What  is  the  primary?  unit  in  United  States  money  1  What  are  tho 
units  of  the  scale  in  passing  from  one  denomination  to  another  ^  How 
docs  this  compare  with  the  scale  in  simple  numbers  1 


S90  API'ENUIX. 

carats  in  the  mixture,  then  the  compound  is  said  to  be  as  many 
carats  fine  as  it  contains  carats  of  the  more  precious  metal,  and  to 
contain  as  much  alloy  as  it  contains  carats  of  the  baser. 

For  example,  if  20  carats  of  gold  be  mixed  with  4  of  silver,  the 
mixture  is  called  gold  of  20  carats  fine,  and  4  parts  alloy. 

4.  Although  the  currency  of  the  United  States  is  in  dollars,  centa 
and  mills,  yet  in  some  of  the  States  the  old  currency  of  pounds, 
shillings  and  pence,  is  still  nominally  preserved. 

In  all  the  States  the  shilling  is  reckoned  at  12  2^^nce,  the  pound 
at  20  shillings,  and  the  dollar  at  100  cents. 

The  following  table  shows  the  number  of  shillings  in  a  dollar, 
the  value  of  £1  in  dollars,  and  the  value  of  $1  in  the  fraction 
of  a  pound : 

In  English  currency,     4s.  6;/.  -  £1  =  $4,84,  and  $1  =  £4.^ 
In  N.  E.,  Va.,  Ky., )     g^_        _  ^^  ^  ^3^      ^^^  ^^  ^  ^    3 

Tenn.,  3 

In  N.  Y.,  Ohio,  N. 

Carolina. 


I    8s.        -  £1  =  $21  and  $1  =:  £    |. 

In  N.  J.,  Pa.,  Del,  I    ^^_  g^^  _  ^^  ^  ^^  ^^^  $1  ^  £    f. 

Md.,  >  •*  " 

In  S.  Carolina  and  Ga.  45.  M.  -  £1  =  $4f ,  and  $1  =  £  J5. 
I  Canad 

Scotia, 


In  Canada  &  Nova  I     ^^^         .  £1  =  $4,       and  $1  =  £    1. 


ENGLISH   MONEY. 

406.    The   units   or  denominations  of  English  money  are 
guineas,  pounds,  shillings,  pence,  and  farthings. 

Notes. — 1.  What  is  the  degree  of  purity  of  the  gold  coins?     Of  the 
silver  coins  ?     Of  the  copper  1 

2.  How  much  pure  gold  in  the  eagle  1     How  much  pure  silver  in  tho 
dollar  \ 

3.  What  is  a  carat  ?     How  are  metals  mixed  by  carats  T 

4.  In  what  denominations  is  money  sometimes  reckoned  in  the  different 
ptates  ? 

IOC.  What  are  the  denominations  of  English  money? 


ENGLISH   MONEY. 


391 


TABLE. 
4  farthings,  marked  far.,  make  1  penny 
12  pence  -         -         -         - 


20  shillings 

- 

1  pound,  c 

21  shillings 

- 

1  guinea. 

far. 

d. 

5. 

4 

=  1 

48 

=  12 

=  1 

960 

=  240 

=  20 

marked     d, 
1  shilling,  -  «, 


£. 


=  1 


TABLE    OF   FOREIGN    COINS    WHOSE   VALUES    ARE    FIXED 

BY   LAW. 


Franc  of  Franco  and  Belgian, 

Florin  of  the  Netherlands, 

Guilder  of  do.    . 

Livre  Tournois  of  France, 

Milrea  of  Portugal,     . 

Milrea  of  Madeira, 

Milrea  of  the  Azores, 

Marc  Banco  of  Hamburg, 

Pound  Sterling  of  Great  Britain, 

Pagoda  of  India, 

Pteal  Vellon  of  Spain, 

Real  Plate  of         do. 

Rupee  Company, 

Rupee  of  Biiiish  India,    . 

liix  Dollar  of  Denmark,     . 

Rix  Dollar  of  Prussia,     . 

Rix  dollar  of  Bremen, 

Ptoublc.  silver,  of  Russia, 

Tale  of  China,    . 

Dollar  of  Sweden  and  Norway, 

Specie  Dollar  of  Denmark, 

Dollar  of  Prussia  and  Northern  States  of  Germany, 

Florin  of  Southern  States  of  Germany, 

Florin  of  Austria  and  city  of  Augsburg,   . 

Lira  of  the  Lombardo  Venetian  Kingdom,     . 

Lira  of  Tuscany,     ...... 

Lira  of  Sardinia, 

Ducat  of  Naples,     ...... 

Ounce  of  Sicily,  ...... 

Pound  of  Nova  Scotia,  New  Brunswick,  Newfound 
laud,  and  Canada,        ..... 


4 
1 


cts. 

40 
40 

12 

00 

83J 

35 

84 

84 

05 

10 

44^ 

44^ 

00 

68^ 

78f 

75 

48 

06 

05 

69 

40 

48^ 

16 

16 

SO 
40 

04 


392 


APPENDIX. 


TABLE    OF    FOREIGN    COIXS    "WHOSE    VALUES    ARE    FIXED 

BY    USAGE. 


Berlin  Rix  Dollar, 
Current  Marc, 
Crown  of  Tuscany, 
Elberfeldt  Rix  Dollar. 
Florin  of  Saxony, 
Bohemia, 
Elberfeldt, 
Prussia,   . 
Trieste, 
Nuremljurg 
Frankfort, 
Basil, 
St.  Gaul, 


(( 
u 
u 
li 
li 
u 
u 
(( 


Creveld, 


Florence  Livre,  . 
Genoa         do., 
Geneva       do.,    . 
Jamaica  Pound, 
Leghorn  Dollar, 
Leghorn  Livre  ((Ji  to 
Livre  of  Catalonia, 
Neufchatel  Livre,    . 
Pezza  of  Leghorn. 
Rhenish  Rix  Dollar, 
Swiss  Livre, 
Scud  a  of  Malta, 
Turkish  Piastre, 


the  dollar), 


ds. 

GH 

28 

05 

69| 

48 

48 

40 

22f 

48 

40 

40 

41 

40 

15 

18| 

21 

00 

90 

o3| 

2(1^ 

90 

60f 

27 

40 

05 


[The  above  Tables  are  taken  from  a  work  on  the  Tariff,  by  E.  D. 
Ogden,  Esq.,  of  the  New  York  Custom  House]. 

Notes. — 1.  The  primary  unit  in  EnglLsh  money  is  1  farthing. 
The  units  of  the  scale,  in  passing  from  farthings  to  pence,  are  4;  ih 
passing  from  pence  to  shillings,  the  units  of  the  scale  are  12 ;  in  pass- 
ing from  shillings  to  pounds,  they  are  20. 

2.  Farthings  are  generally  expressed  in  fractions  of  a  penny.  Thus, 
1/ar.  =  \d. ;  2far.  =  ^l.  ;  Zfur.  --^  \d. 

3.  The  standard  of  the  gold  coin  is  22  parts  of  pure  gold  and  2 
parts  of  copper. 


Note. — 1.  What  are  the  primary  units  of  the  English  currency  1    Name 
the  units  of  the  scale. 


UNITED    STATES    MONEY. 


893 


The  standard  of  silver  coin  is  37  parts  of  pure  silver,  and  3  parts 
of  copper. 

A  pound  of  gold  is  worth  14.2878  times  as  much  as  a  pound  of 
silver.     In  the  copper  coin  24  pence  make  1  pound  avoirdupois. 

By  reading  the  second  table  from  right  to  left,  we  can  see  the  value 
of  any  unit  expressed  in  each  of  the  lower  denominations.  Thus, 
Id.  =  Afar.;  Is.  =  12tZ.  =  48/ar.;  £l  =  205.  =  240rf.  =  960/ar. 

LINEAR   MEASURE. 
407.  This  measure  is  used  to  measure  distances,  lengths, 
breadths,  heights  and  depths. 

TABLE. 


12  inches                 make 

1  foot,             marked    ft. 

3  feet 

- 

1  yard,         -         -         yd. 

51  yards  or  IG^  feet     - 

- 

1  rod,  -         -         -         rd. 

40  rods        _         -         - 

- 

1  furlong,     -         -         fur. 

8  furlongs  or  320  rods 

- 

1  mile,          -         -         mi. 

3  miles      _         -         - 

- 

1  league,      -         -         L. 

69i  statute  miles,  or 

60    geographical  miles,  - 

-] 

1  degree  on  the  )     ,           0 
°                    V   deg.  or    . 

equator,  -      > 

360  degrees 

a  circumference  of  the  earth. 

in.               ft. 

yd. 

rd.            fur.        mi. 

12          =.  1 

36          z=  3          = 

1 

198        =  16L      = 

5* 

=  1 

7920      =  6G0      = 

220 

=  40          =1 

63360    =  5280    =: 

1700 

=  320        =8       =1 

Notes. — 1.  A  fathom  is 

a  length  of  six  feet,  and  is  generally  used 

to  measure  the  depth  of  water. 

2.  A  hand  is  4  inches,  and  is  used  to  measure  the  height  of  horses, 

3.  The  units  of  the  scale,  in  passing  from  inches  to  feet,  are  12  j 
in  passing  from  feet  to  yards,  3  ;  from  yards  to  rods,  5-J;  from  rods  to 
furlongs,  40 ;  and  from  furlongs  to  miles,  8. 


407.  For  what  is  linear  measure  used  1  What  are  its  denominations? 
Repeat  the  table  1  -  What  is  a  fathom  \  What  is  a  hand  ?  What  are-  the 
I'jiits  of  the  scale  in  linear  measure  1 


894  APPESTDIX. 

FOREIGN  MEASURES  OF  LENGTH. 

408.  The  Imperial  jard  of  Great  Britain  is  the  one  from 
■which  ours  is  taken.    Hence,  the  units  of  measure  are  identical. 

"       *  FRENCH  SYSTEM. 

409.  The  base  of  the  new  French  system  of  measures  is  the 
measure  of  the  meridian  of  the  earth,  a  quadrant  of  which  is 
10,000,000  metres,  measured  at  the  temperature  of  32°  Fahr. 
The  muhiples  and  divisions  of  it  are  decimals,  viz. :  1  metre 
=z  10  decimetres  =  100  centimetres  r=  1000  millimetres  = 
8.280899  United  States  feet,  or  39.37079  inches. 

This  relation  enables  us  to  convert  all  measures  in  either 
system  into  the  corresponding  measures  of  the  other. 

Austrian,       1  foot  —  12.448  U.  S.  inches  =  1.03737  foot. 
Prussia!}, 


J 


„, .    ,      ,  I  1  foot  =  12.3G1      "         «       =  1.0300 
Kluneland. 

Swedish,         1  foot  =  11.G90      «         "       =0.974145" 
^  1  foot  —  11.034      "         "       =  0.9195      " 

Spanish,      Y  league  (royal)  =  25000  Span.  ft.  =  4^  miles  "i  >> 
)     "  (common)  =  19800      «  =  ^     ''      \  \ 

CLOTH   MEASURE 
410.  Cloth  measure  is  used  for  measuring  all  kinds  of  cloth, 
ribbons,  and  other  things  sold  by  the  yard. 

TABLE. 
2\  inches,  in.     make     1  nail,     marked         na. 
4    nails         -         -         1  quarter  of  a  yai'd,  qr. 

3  quarters  -         -         1  Ell  Flemish,  E.  Fl. 

4  quarters  -         -         1  yard,         -         -     yd. 

5  quarters  -         -         1  Ell  English,      -     E.  E. 

408.  How  docs   the   Imperial   yard  compare  with  the  standard  in  tlic 
United  States  \ 

409.  What  is  the  unit  of  tlie  French  system  of  measures  ?     How  doci 
the  metre  compare  wilii  our  standard  yard  1 

410.  For  what  is  cloth   measure  used  !     What  arc  its  denominations  1 
Repeat  the  table.     What  arc  the  units  of  the  ucalcc  ? 


SQUAKE   MEASUKE, 


395 


m. 

na. 

qr. 

^.  i^'l. 

yd. 

2i 

=  1 

9 

=  4 

=  1 

27 

=  12 

=  3 

=  1 

36 

=  16 

=  4 

-1^ 

=  1 

45 

=  20 

=  5 

=  lf 

=   U 

KK 


==  1 

Note. — Tho  units  of  the  scale,  iu  this  measure,  are  2^,  4,  3,  J, 
and  -J. 

SQUARE   MEASURE. 

411.  Square  measure  is  used  in  measuring  land,  or  anything 
in  which  length  and  breadth  are  both  considered. 


A  square  is  a  figure  bounded  by  four  equal 
lines  at  right  angles  to  each  other.  Each 
line  is  called  a  side  of  the  square.  If  each 
side  be  one  foot,  the  figure  is  called  a  square 
foot. 


1  Foot. 

o 
o 

I— < 

1  yard  =  3  feet. 


CO 

1 

II 

1— t 

If  the  sides  of  the  square  be  each  one 
yard,  the  square  is  called  a  square  yard. 
In  the  large  square  there  are  nine  small 
squares,  the  sides  of  which  are  each  one 
foot.  Therefore,  the  square  yard  con- 
tains 9  square  feet. 

The  number  of  small  squares  that  is  contained  in  any  large 
square  is  always  equal  to  the  product  of  two  of  the  sides  of 
the  large  square.  As  in  the  figure,  3x3  =  9  square  feet. 
The  number  of  square  inches  contained  in  a  square  foot  is 
equal  to  12  X  12  =  144. 


411.  For  what  is  square  measure  used  ^  What  is  a  square'?  If  each 
side  be  one  foot,  what  is  it  called  1  If  each  side  be  a  yard,  what  is  it 
called  1  How  many  square  feet  does  the  square  j^ard  contain  ?  How  is 
the  number  of  small  squares  contained  in  a  large  square  found  1  Rc()cat 
the  table.     What  are  the  units  of  the  scale  1 


39G  APPENDIX. 

TABLE. 
144  square  inches,  sg.  in.  make  1  square  foot,  Sq.ft. 

9  square  feet  -         -         1  square  yard,  Sq.  yd. 

30^  square  yards        -         -         1  square  rod  or  perch,       P, 
40  square  rods  or  perches  1  rood,    -         -         -         K. 

4  roods  -         -         -         1  acre,    -         -         -         A. 

M. 

A. 


640  acres 

- 

1  square  mile. 

Sq.  in. 

Sq.  ft. 

Sq.  yd.        P.            P. 

144 

=   1 

1296 

=  9 

=  1 

39204 

=   272i 

=  30J   =1 

15G8160 

=   10890 

=z   1210  =40    =  1 

6272G40 

=  43560 

=  4840  =160   =4 

=  1. 

Note. — The  units  of  the  scale  in  this  measure  are  144,  9,  30^.  40, 

and  4. 

SURVEYORS'  MEASURE. 

412.  The  Surveyor's  or  Gunter's  chain  is  generally  used  in 
surveying  land.  It  is  4  poles  or  66  feet  in  length,  and  is 
divided  into  100  links. 

TABLE. 

7^^  inches         make         1  link,  marked       -        -  I. 

4  rods  =66//;,  =  100  links   1  chain,  -         -         -  c. 

80  chains    -         -         -         1  mile,  -         _         -  mi. 

1  square  chain  -         -       16  square  rods  or  perches,  P. 

10  square  chains  -         1  acre,  -         -         -  A. 

Notes. — 1.  Land  is  generally  estimated  in  square  miles,  acres, 

roods,  and  square  rods  or  perches, 

2.  The  units  of  the  scale,  in  this  measure,  arc  7^^^,  4.  80,  1, 

and  10. 

CUBIC   MEASURE. 

413.  Cubic  measure  is  used  for  measuring  stone,  tlmbei, 
eartli,  and  sucli  otlier  things  as  have  the  three  dimensions, 
length,  breadth,  and  thickness. 

412.  What  chain  is  used  in  land  surveying  '>  What  is  its  length  1  How 
is  it  divided  1  Repeat  the  table  1  In  wliat  is  land  generally  estimated  1 
Wiiat  are  the  units  of  the  scale  f 


CUBIC    MKASURK. 


897 


TABLE. 

1728  cubic  inches,  Cu.  in. 

,  make 

1  cubic  foot, 

- 

Cu.ft. 

27  cubic  feet   - 

- 

1  cubic  yard, 

- 

Cu.yd. 

40  feet  of  round  or 
50  feet  of  hewn  timber, 

} 

1  ton. 

- 

T, 

42  solid  feet    - 

- 

1  ton  of  shipping, 

T. 

8  cord  feet,  or  ) 
128  cubic  feet     1 

- 

1  cord. 

- 

C. 

241  cubic  feet  of  stone 

- 

1  perch,    - 

- 

P. 

C3 


CO 


Notes. — 1.  A  cord  of  -wood  is  a  pile  4  feet  wide,  4  feet  hish,  and 

8  feet  long, 

2.  A  cord  foot  is  1  foot  in  length  of  the  pile  which  makes  a  cord. 

3.  A  CUBE  is  a  figure  bounded  by  six  equal  squares,  called  faces  ; 
the  sides  of  the  squares  are  called  edges. 

4.  A  cubic  foot  is  a  cube,  each  of  whose  faces  is  a  square  foot ;  its 
edges  are  each  1  foot. 

5.  A  cubic  yard  is  a  cube,  each  of  whose 
edges  is  1  yard. 

6.  The  base  of  a  cube  is  the  face  on 
which  it  stands.  If  the  edge  of  the  cube 
is  one  yard,  it  will  contain  3X3=9 
square  feet*  therefore,  9  cubic  feet  can 
be  placed  on  the  base,  and  hence,  if  the 
solid  were  1  foot  thick,  it  would  contain 

9  cubic  feet ;  if  it  were  2  feet  thick  it  would  contain  2  tiers  of  cubes, 
or  18  cubic  feetj  if  it  were  3  feet  thick,  it  would  contain  27  cubic 
feet ;  hence, 

The  contents  of  such  a  figure  are  found  by  multiplying  the  length, 
breadth,  and  thickness  together. 

7.  A  ton  of  round  timber,  when  square,  is  supposed  to  produce  40 
cubic  feet :  hence,  one-fifthjs  lost  by  squaring. 


3  feet  =  1  yard. 


413.  For  what  is  cubic  measure  used!     What  are  its  denominations? 

What  is  a  conl  of  wood  1     AVhat  is  a  cord  foot  ?     What  is  a  cube'    What 

is  !i  cubic  foot  1     What  is  a  cubic  yard  ?     How  many  cubic  feet  in  a  cubic 

yard  1     What  are  the  contents  of  a  volume  equal  to  ?     Repeat  the  table. 

What  are  tlic  units  of  the  scale  1 

18 


898 


APPEISTPIX. 


WINE   MEASURE. 

414.  "Wine  measure  is 

used  for  measuring  all  li 

[quids  except 

ale,  beer  and  milk. 

TABLE. 

4  gills,  gi. 

make 

1  pint,         marked 

pt. 

2  pints 

- 

1  quart, 

qt. 

4  quarts     - 

- 

1  gallon, 

gal. 

31 1  gallons   - 

- 

1  barrel. 

bar.  or  hbl 

42  gallons   - 

- 

1  tierce. 

tier. 

63  gallons    - 

- 

1  hogshead, 

hhd. 

2  hogsheads 

- 

1  pipe, 

2n. 

2  pipes  or  4  hogsheads 

1  tun. 

tun. 

gi.         pL 

qt. 

gal.      bar.    tier.     hhd. 

.    pi.     tun. 

4         =1 

8         =2 

=  1 

32       =8 

=  4 

=  1 

1008  =  252 

=  126       : 

=  311  ^1 

1344  =  336 

=  168       : 

=  42             =1 

2016  ^  504 

=  252        : 

=  63             =  11  =  1 

4032   =  1008 

=.504       : 

=  126           =3     =2 

=  1 

8064  z^  2016 

zzz   1008     : 

=  252           =6=4 

=  2=1. 

Notes.— 1.  The  standard  unit,  or  gallon  of  wine  measure,  in  the 
United  States,  contains  2.31  cubic  inches,  and  hence,  is  equal  to  the 
weight,  avoirdupois,  of  8.339  cubic  inches  of  distilled  water,  very 
nearly. 

2.  The  English  Imperial  wine  gallon  contains  277.274  cubic 
inches,  and  hence,  is  equal  to  1.2  times  the  wine  gallon  of  the  United 
States. 

BEER   MEASURE. 
415.  Beer  measure  is  used  for  measuring  ale,  beer,  and  milk. 


414.  What  is  measured  by  wine  measure  1  AVliat  arc  its  dcrominations  1 
Rcpcr.t  the  table.  What  are  the  units  of  the  scale  1  What  is  a  standard 
wine  gallon  1 

415.  For  what  is  beer  measure  used?  What  are  its  denominations  1 
Repeat  the  taiilc      What  arc  the  units  of  the  scale  1 


DKY    MEASUKE.  399 


TABLE. 

2  pints,  pt.         make  1  quart,         marked 

qt. 

4  quarts     - 

1  gallon,    - 

gal. 

36  gallons    - 

1  barrel,    - 

bar. 

54  gallons    - 

1  hogshead. 

hhd. 

pt.                  qt. 

gal.           bar. 

hhd. 

2              =1 

8            =4 

=     1 

288         =  144 

=  36         =1 

432         =  216 

=  54         =11 

=  1. 

Notes. — 1.  The  standard  gallon,  heer  measure,  contains  282  cubic 
inches,  and  hence,  is  equal  to  the  weight  of  10.1799  cubic  inches  of 
distilled  rain  water. 

2.  Milk  in  many  places  is  sold  by  wine  measure. 

DRY    MEASURE. 
416.  Diy  measure  is  used  in  measuring  all  dry  articles,  such 
as  grain,  fruit,  salt,  coal,  &;c. 

TABLE. 

2  pints,  pt.      make     1  quart,     marked  qi. 

8  quarts    -         -         1  peck,      -         -  pk. 

4  pecks     -         -         1  bushel,  -         -  biu 

36  bushels  -         1  chaldron,         -  ch. 

pt.  qt.  pk,  hu.  ch. 

2  =1 

16  =8  =1 

64  =32  =4  =1 

2304  =  1152  =  144         =36  =1. 

Notes. — 1.  The  standard  bushel  of  the  United  States  is  the  Win" 
chester  bushel  of  England.  It  is  a  circular  measure  18-J  inches  in 
diameter  and  8  inches  deep,  and  contains  2150.4  cubic  inches,  nearly. 
It  contains  77.627413  pounds  avoirdupois  of  distilled  water. 

2.  A  gallon,  dry  measure,  contains  268.8  cubic  inches. 

416.  What  articles  are  measured  by  dry  measure  1  What  are  its  de- 
nominations 1  Repeat  the  table.  What  are  the  units  of  the  scale  1  What 
ir  the  standard  bushel !     What  are  the  contents  of  a  ffallon '' 


400  APPENDIX. 

3.  Wine  measure,  Beer  measure,  and  Dry  measure,  and  all  mea- 
sures of  volume,  differ  from  the  cubic  measure  only  in  the  unit  which 
is  used  as  a  standard. 

AVOIRDUPOIS    WEIGHT. 

417.  By  this  weight  all  coarse  articles  are  weighed,  such  as 

bay,  grain,  chandlers'  wares,  and  all  metals  except  gold  and 

silver. 

TABLE. 

16  di'ams,  dr.  make  1  ounce,  marked  oz. 
16  ounces  -  -  1  pound,  -  -  Ih. 
25  pounds  -         1  quarter,    -         -         qr. 

4  quarters  -         1  hundred  weight,         cwt. 

20  hundred  weight       1  ton,  -         -         T. 

dr.  oz.  lb.  qr,  cwt.  T. 

16     =  1 

256     nr  16  =  1 

6400     r=:  400  =25  =1 

25600     =  1600  =100         =4         =1 

512000     =  32000  =  2000       =80       =20         =1. 

Notes. — 1.  The  standard  avoirdupois  pound  is  the  weight  of  27.701 5 
cubic  inches  of  distilled  water;  and  hence,  1  cubic  foot  weighs  1000 
ounces,  very  nearly. 

2.  By  the  old  method  of  weighing,  adopted  from  the  English  sys- 
tem, 112  pounds  were  reckoned  for  a  hundred  weight.  But  now,  the 
laws  of  most  of  the  States,  as  well  as  general  usage,  fix  the  hundred 
weight  at  100  pounds. 

3.  The  units  of  the  scale,  in  passing  from  drams  to  ounces^  are  16  ; 
from  ounces  to  pounds,  16,  from  pounds  to  quarters,  25  j  from  quar- 
ters to  hundreds,  4  ;  and  from  liundreds  to  tons,  20. 


417.  For  what  is  avoirdupois  weight  used  1  How  is  the  table  to  be 
read  ^  How  can  you  determine,  from  the  second  table,  the  value  of  any 
unit  in  units  of  the  lower  denominations  1 

NoTF.s. — 1.   What  is  tlie  standard  avoirdupois  pound  1 

2.  What  is  a  hundred  weigiit  by  the  English  method  ]  What  is  a  hun- 
dred weight  by  the  United  States  method  1 

3.  Name  the  units  of  the  scale  in  passing  from  one  denominatijii  to 
another. 


TROY    WEIGHT.  401 

TROY  WEIGHT. 
418.  Gold,  silver,  jewels,  and  liquors,  are  weighed  by  Troy 

wciglit. 

TABLE. 


24  grains,  gr. 

make 

:    1 

pennyweight. 

marked  pu 

20  pennyweig 

hts     - 

1 

ounce, 

oz. 

12  ounces 

- 

1 

pound. 

lb. 

gr. 

pwt. 

oz. 

lb. 

24 

=  1 

480 

=  20 

=  1 

57G0 

=  240 

=  12 

=  1. 

Notes. — 1.  The  standard  Troy  pound  is  the  weight  of  22.794377 
cubic  inches  of  distilled  "water.  Hence,  it  is  less  tlian  the  pound 
avoirdupois 

2.  7000     troy  grains  =       1   pound  avoirdupois. 

175     troy  pounds  =  144  pounds         " 
175     troy  ounces  =  192  ounces  " 

437-^  troy  grains  =      1  ounce  " 

3.  The  Troy  pound  being  the  one  deposited  in  the  mint  at  Phila- 
delphia, is  generally  regarded  as  the  standard  of  weight. 

4.  The  units  of  the  scale  are  24,  20,  and  12. 

APOTHECARIES'    WEIGHT. 
419.  This  weight  is  used  by  apothecaries  and  physicians  in 
mixing  their  medicines.     But  medicines  are  generally  sold,  in 
the  quantity,  by  avoirdupois  weight. 

TABLE. 

20  grains,  gr.         make     1  scruple,  marked  9 . 

3  scruples      .         -         1  dram,     -         -  3 . 

8  drams  -         -         1  ounce,    -         -  5 . 

12  ounces         -    -^    -         1  pound,    -         -  Ife-. 

418.  What  articles  are  weighed  by  Troy  weight"!  What  are  its  denomi- 
nations 1  Repeat  the  table.  What  is  the  standard  Troy  pound  ?  What 
are  the  units  of  the  scale,  in  passing  from  one  unit  to  another  1 

419.  What  is  the  use  of  Apothecaries' weight  1  What  are  its  denomi- 
nations 1  Repeat  the  table.  What  are  the  values  of  the  pound  and  ounce  1 

What  are  the  units  of  the  scale,  in  passing  from  one  unit  to  another  1 

18 


402 


APPENDIX 

gr- 

9 

3 

20 

=  1 

60 

=  3 

r=   1 

480 

=  24 

=  8 

ft 


=  1 

5760         =288         =96  =12  =1 

Notes. — 1.  The  pound  and  ounce  are  the  same  as  the  pound  and 
ounce  in  Troy  weight. 

2.  The  units  of  the  scale,  in  passing  from  grains  to  scruples,  are 
20  ;  in  passing  from  scruples  to  drams,  3  :  from  drams  to  ounces,  8 ; 
and  from  ounces  to  pounds,  12. 

NEW   FRENCH    SYSTEM. 

420.  The  basis  of  this  system  of  weights  is  the  weight  in 
vacuo  of  a  cubic  decimetre  of  distilled  water.  This  weight  is 
called  a  kilogramme,  and  is  the  unit  of  the  French  system.  It 
is  equal  to  2.204737  pounds  avoirdupois.  The  other  denomi- 
nations are  as  follows  : 

100  kilogrammes  =  1  quintal ;  10  quintals  =  1  ton  sea  water; 
1  gi-amme  =  10  hectogrammes ;  1  hectogramme  =  10  deco- 
grammes  ;  1  decogramme  =  10  grammes  ;  1  gramme  =  10  de- 
cigrammes ;  1  decigramme  =  10  centigrammes ;  1  centigramme 
=  10  milligrammes. 

COMPARISON   OP   WEIGHTS. 

English^  1  pound  =  1.000936  pounds  avoirdupois. 

French,  1  kilogramme  =  2.204737       "  " 

Spafiish,  1  pound  =  1.0152  "  " 

Swedish,  1  pound  =  0.9376  "  « 

Austrian,  1  pound  =  1.2351  "  " 

Prussian,  1  pound  =  1.0333  «  « 

MEASURE    OF   TIME. 

421.  Time  is  a  part  of  duration.  The  time  in  which  the 
earth  revolves  on  its  axis  is  called  a  daij.  The  time  in  which 
it  goes  round  the  sun  is  called  a  solar  year.  Time  is  divided 
into  parts  according  to  the  following 


DA 

xr.s.                                         40y 

TABLE. 

60  seconds,  sec. 

make     1   minute,         marked     m. 

60  minutes 

1   hour,           -          -         hr. 

24  hours 

1   day,            -          -         da. 

7  davs 

1   week,         -         -        wk. 

4  weeks 

1   month,  nearly,  -        mo. 

12  calendar  m'ths  = 

365(/rt.   1  Julian  or  civil  year,    yr. 

866  days 

1   leap  year. 

100  years 

1  century,     -         -      cent. 

see.                  VI. 

hr.         da.              xok.         yr. 

60                =1 

3600            =60 

=  1 

&6400          =1440 

=  24        =1 

604800       =100S0"      =168     =7          =1 

31536000  =525600     ^8760   =365     =521  =1. 

The  year  is  divided  into  12  calendar  months  : 

No.                        No. 

days. 

No.                         No.  days. 

1st.  January,       -     - 

31 

7th.  July,      -     -     -     -     31 

2d.    February,    -     - 

28 

8th.  August,       -     -     -     31 

3d.    March,    -     -     - 

31 

9th.  September,      -     -     30 

4th.  April,      -     -     - 

30 

10th.  October,      -     -     -     31 

5th.  May,       -     -     - 

31 

11th.  November,-     -     -     30 

6th.  June,       -     -     - 

30 

12th.  December,       -     -     31 

The  number  of  days 

in  each  month  may  be  remembered  by 

the  following : 

Thirty  days  hath  September, 

April,  June,  and  November  ; 

All  the  rest  have  thirty-one. 

Excepting  February,  twenty-eight  alone. 

421.  What  are  the  denominations  of  time  1  How  long  is  a  year  1  How 
many  days  in  a  common  year  1  How  many  days  in  a  Leap  year  ]  How 
many  calendar  months  in  a  year  '\  Name  each,  and  its  number,  and  the 
number  of  days  in  each.  How  many  days  has  February  in  the  leap  year  \ 
How  do  you  remember  which  of  the  months  have  30  days,  and  which  31 1 

jVoTE. — 1    How  are  the  centuries  numbered  1     How  are  the  years  nuin 
eared  1     The  days  1     The  hours  1 


404  APPENDIX. 

Notes. — 1.  Days  are  numbered  in  each  month  fiom  the  first  day 
of  the  month. 

2.  Montlis  are  numbered  from  January  to  December. 

3.  The  centuries  are  numbered  from  the  beginning  of  the  Christian 
Era.  The  year  30,  for  example,  at  its  commencement,  was  called 
the  30th  year  of  the  first  century,  though  neither  the  century  nor  the 
year  had  elapsed.  Thus,  June  2d,  1856,  was  the  6th  month  of  the 
56th  year  of  the  19lh  century. 

4.  The  civil  day  begins  and  ends  at  12  o'clock  at  night.  In  the 
civil  day,  the  hours  are  reckoned  from  that  time. 

DATES. 

1.  The  length  of  the  solar  year  is  365da.  5hr.  48m.  4S.tcc..  very 
nearly.  It  is  desirable  to  have  the  periods  and  dates  of  the  civil 
year  correspond  to  those  of  the  solar  year  ;  else,  the  summer  months 
of  the  one  would  in  time  become  the  winter  months  of  the  other, 
thereby  producing  great  confusion  in  dates  and  history. 

2.  The  common  civil  year  is  reckoned  at  365^/a.,  and  the  solar 
year  at  365da.  6hr.  The  6  hours  accumulate  for  4  years  before  they 
are  counted,  when  they  amount  to  1  day,  and  are  added  to  February 
and  the  year  is  called  a  bissextile  or  leap  year. 

3.  The  odd  6  hours  have  been  so  added  that  the  leap  years  occur 
in  those  numbers  which  are  divisible  by  4.  Thus,  1856,  1860,  1864, 
&c.,  ar3  leap  years;  and  when  any  number  is  not  divisible  by  4,  the 
remainder  denotes  how  many  years  have  passed  since  a  leap  year. 

4.  This  method  of  disposing  of  the  fractional  part  of  the  year 
would  be  without  error  if  the  solar  year  were  exactly  365da.  6hr.  in 
length  ;  but  it  is  not;  it  is  only  365(la.  5hr.  48m.  48scc.  long  :  hence, 
tt;C  leap  year  is  reckoned  at  too  much^  and  to  correct  this  error,  every 
centennial  year  is  reckoned  as  a  common  year.  But  this  makes  an 
error  again,  on  the  other  side,  and  every  fourth  centennial  year  the 
day  is  retained.  Thus,  1800  was  not,  and  1900  will  not  be.  reckoned 
a  leap  year  :  the  error  will  then  be  on  the  other  side,  and  2000  will 
be  a  leap  year.  This  disposition  of  the  fractional  part  of  the  year 
cau,scs  the  civil  and  solar  years  to  correspond  very  nearly,  and  indi- 
cates the  following  rule  for  finding  the  leap  years : 

Every  year  ivhick  is  divisible  by  4  is  a  leap  year,  unless  it  is 
a  centennial  year,  and  then  it  is  not  a  leap  year  tmless  the  num' 
her  of  the  century  is  also  divisible  by  4. 

5.  Tlie  registration  of  the  days,  by  reckoning  the  civil    year   at 


jnSCKIXANKOUS    TABLES.  405 

365da.,  was  established  by  the  Roman  Emperor,  Julius  Ca5sar,  and 
hence  Ihis  period  is  sometimes  called  the  Julian  year. 

The  error,  arising  from  the  fractional  part,  continued  to  increase 
until  15S2.  when  it  amounted  to  10  days;  that  is,  as  the  year  had 
been  reckoned  too  long  the  number  of  days  had  been  too  feu\  and  the 
count,  in  the  civil  year,  was  behind  the  count  in  the  solar  year. 

In  this  year,  (1582),  Pope  Gregory  decreed  the  4th  day  of  October 
to  be  called  the  14th,  and  this  brought  the  civil  and  the  solar  years 
to^ether.  The  new  calendar  is  sometimes  called  the  Gregorian 
Calendar. 

6.  The  method  of  dating  by  the  old  count,  is  called  Qld  Style ; 
and  by  the  new,  New  Style.  The  difference  is  now  12  days.  la 
Russia,  they  still  use  the  old  style;  hence,  their  dates  are  12  days 
behind  ours.     Their  4th  of  January  is  our  16th. 

CIRCULAR   MEASURE. 

422.  Circular  measure  is  used  in  estimating  latitude  and 
longitude,  in  measuring  the  motions  of  the  heavenly  bodies,  and 
also  in  measuring  angles. 

The  circumference  of  every  circle  is  supposed  to  be  divided 
into  360  equal  parts,  called  degrees.  Each  degree  is  divided  into 
60  minutes,  and  each  minute  into  60  seconds. 


TABLE. 

60  seconds  "         make 

1  minute. 

marked 

• 

60  minutes 

1  degree. 

- 

O 

• 

30  degrees 

1  sign,     - 

- 

s. 

12  signs  or  360° 

1  circle,  - 

- 

c. 

//               1 

o 

s. 

c. 

60       =  1 

3600       =  60 

=  1 

lOSOOO       r=  1800 

=  30 

=  1 

1296000       =  21600 

=  360 

=  12 

=  1. 

422.   For  what  is  circular  measure  used  !     How  is  every  circle  supposed 
lu  be  divided  1     Repeat  the  table. 


406 


APPKNDIX, 


MISCELLANEOUS    TABLE 

12  units,  or  things  make 

12  dozen    -         -         _         . 
12  gross,  or  144  dozen 


20  things  -         -  .  . 

100  pounds  -  -  _ 

196  pounds  -  _  _ 

200  pounds 

18  inches  -         -  -  - 

22  inches,  nearly 

14  pounds  of  iron  or  lead  - 

21^  stones  -         -  -  _ 

8  pigs      -         .  -  . 


1  dozen. 

1  gross. 

1  great  gross. 

1  score. 

1  quintal  of  fish. 

1  barrel  of  flour. 

1  barrel  of  pork. 

1  cubit. 

1  sacred  cubit. 

1  stone. 

1  fother. 


BOOKS   AND    PAPER. 

The  terras,  folio,  quarto,  octavo,  duodecimo,  Sec,  indicate  the 
number  of  leaves  into  which  a  sheet  of  paper  is  folded. 


A  sheet  folded 
A  sheet  folded 
A  sheet  folded 
A  sheet  folded 
A  sheet  folded 
A  sheet  folded 
A  sheet  folded 
A  sheet  folded 


n    2  leaves  is  called  a  folio. 

.1    4  leaves         "       a  quarto,  or  4to. 


n  8  leaves 
n  12  leaves 
n  16  leaves 
in  18  leaves 
n  24  leaves 
n  32  leaves 


a 
ii 
ii 
a 

a 


24  sheets  of  paper 
20  quires  - 

2  reams  - 

5  bundles 


make 


an  octavo,  or  8vo. 

a  12mo. 

a  16mo. 

an  18mo. 

a  24mo. 

a  32mo. 

1  quire. 
1  ream. 
1  bundle. 
1  bale. 


ANSWERS. 


r. 

EX. 

1 

ANS. 

EX. 

"8 

A^S.              1 

EX. 

15 

ANS. 

16. 

XVI. 

DCCLI. 

DCCCCLVII. 

16. 

2 

XIV. 

9 

MLX. 

16 

MCCVI. 

IG. 

O 

XVIII. 

01 

MMXCI. 

17 

CCCCXCV. 

16. 

4 

LXIX. 

11 

DLXIX. 

18 

DCCLV. 

16. 

5 

LXXVIII. 

12 

DCCXLV 

19 

MDCCCXLVII. 

16. 

6 

CXV. 

13 

DCCCCLXI 

20 

MMDXX. 

16. 

7 

CCCCIX. 

14 

DCXCIX. 

19. 

1          7 

3        9000        11 

5 

1    961 

6       7408 

19. 

2        80 

4 

93 

- 

20. 

7 

897.021 

20. 

8 

86.029,430 

20. 

9 

4,328,021,063 

20. 

10 

967,040,932 

20. 

11 

30,430,208,123 

20. 

12 

360,030,702,010 

20. 

13 

5,800,006,000,812 

14 
15 
16 

17 
18 
19 
20 


75,605,070,905,008 

904,000.800,200,720 

6,000.900,704,098,020 

80,510,006,040,900,040,900 

6,050,900,001 

987,654,321.012,345,678 

208,104,111,001,111,111 


23. 
23. 
23. 
23. 
23^ 

24. 
24. 
24. 
24. 
24. 
24. 
24. 


2 

o 
O 

4 
5 


621 

6 

5.702 

7 

8,001 

8 

10,406 

9 

65,029 

10 

40,000,241 

59,000,310 

12,111 

300,001,006 

69,003,000,211 


11 

12 

13 

14 

15 

16 

17 

47.000,069,000,465,207 

800. 000,'000,000,429, 006,009 

95,000,000,000,000,089,089,306 

6,000,000,451,065,047,104 

999,065,841,411 

470,040,000.000,000,000,000,000,004,006,204 

65.000,800,000.750,751,975,310 


3L 

33. 
33. 

34. 
34. 
34. 
34. 
34. 


2;   7 


1 


\  42600  ; 
(426000 


36b60 
^8,75 


10|8996 


ll|i;i  125.  &c^.    Ifar. 


12 
13 
14 
15 
16 


4  5445/5. 
IT.  lAc'ct.  \qr.  20/6. 


262155-r5. 


\22lb.  2oz.  ISpwt.  9gr. 
29:]ii2gr. 


17  mib.  4  §   3  5   2  9   7gr. 
I8\2i9i?i. 

1600rJ.  SSOOyds. 

26400/?.  316800m 
20  r/oi/d.  2ft.  6in. 


,9 


403 


ANSWERS. 


P. 

EX 

21 

ANS. 

EX. 

30 

ANS. 

34. 

6sq.  t/d.   2v7.  ft. 

78  E.E. 

\qr. 

34. 

22 

2 A.  OH.   35P. 

31 

1008^^. 

34. 

23 

^5A.  Q)sq.  Oh. 

32 

1  Z)hhd. 

I 

34. 

21 

56SP. 

33 

•i02\pt. 

34. 

25 

967  GSOcu.  in. 

34 

\29har. 

34. 

26 

o968cu.ft.. 

35 

1984;)^. 

34. 

27 

iiOcords. 

36 

Oc/i.  o2hu.  o'ph. 

Iqt. 

34. 

28 

25  \  2  )ta. 

37 

6311385C 

)sec. 

34. 

29 

lUj/d.. 

38 

Smo.  2ick. 

38. 

1 

182630 

--J 

1 

395873 

13 

32921 

38. 

2 

87539 

8 

30534 

14 

185876 

38. 

110526 

9 

74716 

15 

93684 

38. 

4 

79165 

10 

29909 

16 

34289 

38. 

5 

73285 

11 

74022 

17 

243972 

38. 

r 

41-18907 

12 

833516 

J 

39. 

18 

39. 

19 

39. 

20 

39. 

21 

39. 

22 

39. 

23 

39. 

24 

39. 

25 

39. 

j26 

§991,546. 

•^85,465. 

4770,560, 

^525.892. 

f;9638.495. 

£223  2s.  5d.     If. 

1295/6.    lOoz.  2jnct. 

4531b    9  !    3  3 

2ciol.ogr.8lb.8oz.  5d>: 


27j 

2s: 

29 

3o: 

31 

32 

..  .■> 
oo 

34 


43  T.  2cict.   Qqr.  lib. 
312i/d.   Qqr.  2da. 
2d\  E.  E.   \qr.  3na. 
143jL.   2)7ii.   6  fur. 
4 fur.  -ird.   Oi/d.    \ft. 
i22A.   \R.  IP. 
2224  T.  Ohlid.  5gal. 
oqt. 


Tin. 


19  gal. 


40. 
40. 
40. 
40. 
40. 


35 
36 
37 
38 
39 


2o0chal.  2bbu.  opk.  Aqt. 

823  ?/r.    I0??i0. 

9Q4da.   \8hr.   Imi. 

2  2\lAcivt.lqr.2Qlb.  l5oz. 

,23592550. 


40 
41 
42 
43 
44 


-S137915910. 

68056. 

12 Imi.  4fiir.  8rd.  5ft. 

519190. 

1124749. 


41. 

i 

15 

$22,009. 

1 

50  ;  29026. 

41. 

46 

$27,740. 

51   8209.75. 

41. 

47 

2tMn.  2hhd.  29gal. 

2qt.  Ojn. 

52  26326421. 

41. 

48 

§20308675. 

53   29714. 

41. 

49 

30569S53. 

54   50110025. 

42. 

55 

5980^512 

60 

Alb.  ooz.  (jjnct. 

42. 

56 

2T.  Acict.  2qr.  lib. 

61 

1053420 

42. 

57 

205  acre  a. 

62 

1089507 

42. 

OH 

S75002,295 

63 

32341 

42. 

59 

$7425 

— 

ANSWERS. 


409 


P. 

EX. 

ANS. 

e:c. 

ANS. 

43. 
43. 
43. 
43. 
43. 

64 

65 
CG 
67 
68 

$27131,23 

$28,105 
39//f/.  }qr. 

$180,825 
$35068,807 

69 
70 
71 
72 

481125 
66585383 

$1019,10 
$33800 

44. 
44. 

73 

74 

380  bu.  \2^k. 
$458,342 

75 
76 

£57  lis.  2d.  ofar. 

58G0 

47. 
47. 
47. 
47. 

1 
2 

o 
O 

4 

363296 

56579 

733071 

17  11 0O7 

5 

6 

7 
8 

41923288  rods. 

$7838180 

106026  vtills. 
4-^01 /)/v.. 

9 
10 
11 

62786/;^. 
198621115m. 
3591 75765  b;>. 

48. 
48. 
4S. 
48. 
48. 
48. 
48. 
48. 
48. 
48. 
48. 
48. 
48. 
48. 


12 

13 
14 

15 
16 
17 

18 

19 

120 

21 

2-2 

23 

;25 


4199675  cords. 

S87877S<ral. 

99999977//>. 

$3-143.641 

$806,384 

$4853673,758 

£14   18s.  3^/.   l//r. 

3  T.  8cioL  2qr.  7 lb. 

i\7 yds.  2qr.  \na. 

odL.   \mi.  ofar.  28rd. 

8t,un  Ihhd.  bogal.  2>qt. 

89  A  2R.  37 P. 

976bu.  Ipk.  6qt. 

124  cords  5Sft..  522in. 


26 
27 
28 
29 
30 
31 


36 
37 
3.8 


25  E.  E.  \cjr.  3;i«. 
79fe  10  !    6  3 
12  3    4  5  2  9 
124  £.  E.  3qr.  3}ia. 
9GE.  F.  Iqr.  Ina. 
12  T.   newt.  3qr. 
2cwL  2/r.  22/6. 
G9qr.  2lb.  lioz. 
l34//>.  l4oz.  \odr. 
10 A  2R.  18P. 
37 A.  2R.  34P 
] 47 da.  2\hr.  5(Jmi. 
o2kr.  50)11.  54isec. 


49.1 

39 

49.! 

40 

49.! 

41 

49.1 

42 

49.! 

43 

49. 

4^ 

49. 

45 

$8759,625 

183666662 

(■>?/;•.  9mo.  3ick.  Wda. 

8b  fe  0  !   6  3 

$8,20 

$39,808 

$10,626 


46 
47 
48 
19 
50 
51 
52 


£121  175, 

6///.  Qmo. 

6353870 

5747 

$6020 

25712808,91 

$36190 


Od.  \far. 
Owk.  Gda. 


9hr.  2tni 


odJ 
50.1 
50.1 
50.' 
5,). 
5^ 

51. 
51 


Ob 

54 
■55 
56 
57 


GS3021 

107445034 

6274 

4  T.  3civL  2qr. 

£19   19.y.  2d. 


23lb. 
ydr. 


58\2299mi.  2far.  4.rd. 


59 
60 
61 
62 
63 
64 


>;  19 9,625  lost. 

S175,875 

$3,25 

19987563 

2899248 

i;73675 


'    65 

66 


22815 
$198,625 


67 
68 


80///-.  87710.  Oda.  3hr.  30m. 
$655,125 


410 


ANSWERS. 


p-i 

EX. 

ANS. 

EX. 

AXS. 

51. 

G9 

249?//-.  Imo.  Wda. 

73 

§7398 

51. 

70 

17877 

74 

-$23G0 

5!. 

71 

S731U756 

75 

526 

51. 

72 

$62727794 

76 

6274 

77 

78 
79 
80 
81 

82 


$356.35  gained. 

3J..  2ii.  39P. 

41  cords  5  cord  feet. 

$3280,105 

$44161,987 

$14352,50 


83 
84 

85 
86 
87 


2?/A  8?reo.   I9(^a. 

iOgal.  2qt.   Ipi. 

50062 

15550 

12°   23'  53'' 


88 

$161,175  gained. 

,,  90 

89 

2271707 

i:  91 

32y(^.  O^-;-.  2«a. 
£950   2s.  8d. 


1 
2 

»"> 
o 

4 

5 

6 

7 

8 

9 

10 

11 

12 


6776368 

68653214 

3422454 

1952883 

4354224 

1028.540646 

24668698404 

$70,84 

$12517,764 

>;96 1662,960 

201638228149 

4281770760 

174809600 


141301144560000 
15|610071000 

16  14783518400 

17  £81   6s.   8d. 

1 8  24  T.  TcivL  oqr. 
W\UQyd.   \fl.   2>in. 
20ill4°  26'   15" 
2\\5i6hhd.  Igal.  2qt. 
22 [598  E.  F. 

23 1 865  T.  11  cm;^.  Syr.   20/5. 
24]320y?-.  2mo.  Oiv/c.  Ida.  lohr.  I2f)u 
2514896 


Opt. 


26 
27 
26 
29 
30 
31 
32 


34 
35 
36 
37 

38^ 


234048 

4482566 

314986464 

320021195962 

556321146761 

1747125213301 

23246S4880333 

71109696192112 

90012355857332 

549600 

670460; 6704600 

570  1900; 57049000 

j  498U496000  ; 

1  49804960000 


,59 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 


9072040000  ;  907204000000 

74040900;  740409000 

67493600;  67493600000 

129359360000 

13729103000000 

664763206000000 

879923S229600000 

25264260 17908695000000 

10936893 68 145U8437S777040 

16714410677359581583737 

$61975 

3240WU. 

§2097 

\33t/d.  3qr.  2na, 


ANSWKKS. 


411 


p. 

EX.                            ANS. 

EX. 

54 

ANS. 

62. 

Do    £3  1 95.  4d.  2far. 

11031,68 

62. 

55 

62.  i 

56 

62.1 

57 

62. 

58 

62.1 

59 

62.' 

60j 

$15 

$506,88 

$6336 

$5545 

$16763832 

496mi.  \fur.  24:rd. 


61' 
62 
63 
64 
65 
66 


63.  i 

67 

63. 

68 

63. 

69 

63. 

70 

63. 

71 

63. 

72 

63. 

'^O 

63. 

JO 

146484  yards. 

427816  barrels. 

$84,26 

$16875,60 

2T.  \8cw(.  Iqr.  2\lb. 

$971,04 

461  barrels  left. 

$1315  cost. 


74 
75 
76 
77 

78 


<oda.  6hr.  37m. 

QoldoJlars. 

$24,375 

SQSmiles, 

71fe  2§    3  7  0  9l2yrs. 

411/Ow.  Vpk.  Oqt. 


$1417 

$65962788,75 
750  days. 
$13500 
$243 


64. 
64. 
64. 
64. 
64^ 
65. 
65, 
65 

66, 


79 

80 
81 
82 
83 


$4770,755 
$61 
$672 
$11914 
286yr.  9mo. 


84 
85 
86 
87 


SArd.  Uft. 

50 

24:Cords. 

$92 


88 
89 
90 


216  men. 
$149,25 
37816  tons. 


91  :  $34.88 


92 
93 
94 


Abar.  32ya/.  3-7/. 
669hhd.  40(/al.  2qt. 
$13650000 


95 
90 
97 


$202,50 
$21,475 
$927,35 


71. 
71. 
71. 

72^ 
72. 
72. 

72. 
72. 
72. 
72. 
72. 
72. 
72. 
72, 
72 


98|  $18844,01  II  99|  $132,9*35  ||  100  |  £175  18s.  6d 


G579 

36842 

269368 


4 
5 


275155 
794S312 


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7 


1147187 
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£15  195.  9d. 
\A  OB.  33 P. 
9i/d.  2jr.  Ina. 
$79,3445 

$209,728 

$66862,18 

153114091-1 

237132 

177242 


9 
10 
11 
12 
13 
14 
15 
16 
17:68 


18 
19 


44670 
071  4 


20 
21 
22 
23 
24 
25 
26 
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29 
30 
31 


$17,4512 

32 

$3,842^-11^ 

33 

$1,125 

34 

$0,375 

35 

$0,81 

36 

$5,01 

37 

$52,88 

38 

9 

39 

95 

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412 


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P. 

EX. 

44 

72. 

72. 

45 

72. 

46 

72. 

1  47 

AX?. 


17^1.  3B.  IP. 
Ida.  12hr.  olm.  SOsec. 
3omi.  Ofiir.  29rd. 
4-%a/.  S^^^qt. 


KX. 

48 
49 
50 

AXS. 


2hi(sh.  Opk.  Iqt. 

?=2o,2o 

2^.  Ad. 


73. 
73, 
73. 
73. 
73. 
73. 


51 
52 
53 
54 
55 
56 


22mi.  \fitr.  8rd. 
31 6 J.  IR.SoP. 

$27,  397  + 
98765 
$11250 
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57i$4.75 


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JO 


59,757  ISByLS, 
60$1,625 


61 


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62S00008 


63J47  ririffs. 
64\lT.  Ucu't.  2qr. 
65  45c-.  /■/.  9954c.  in. 
66b0liff/on.s-.' 
67    44242f^-oZ5. 


74. 

2 

74. 

3 

74. 

4 

74. 

5 

7175 
4600 
168525 
76850 


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75. 
75. 
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4800      1 

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2 

5950 

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3 

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559750661 

493574661 


3558504001 
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2  219917G000 

3  242601500 

4  17573500 


76. 
76. 
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2 

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254 

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140848 

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242172 
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77.  j 
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82.     1  1  $121,615    ]  2     $67,50   |  3  |  $118,9145 

83.|     1  1  $3,024       II  2     $12,8915     3  |  $9,198  ||  4|  $18,22765 

85.     1 

3  5°  13'  difT  in  long. 

2 

lA?-.  2//^  8i-cc.  P.M. 

86.     3 
86.     4 

13°  23' 
4w.  36.SYC. 

5 
6 

8/^r.  1 2m.  A.  M. 
10°  34' 

ANSWERS. 


413 


p. 

EX. 

7 

ANS.                                      I 

1 

ANS. 

86. 

35°  11' 

$128 

86. 

8 

P5°  48'  West, 

2 

2bu.  \pk. 

86. 

\  10 /.r.  17m.  4Ssfc.P  M. 

3  , 

$53.28 

86. 

9 

120° 

4 

32  barrels. 

86. 

10 

1/ir.  2m.  2G.'?«c.  Fast. 

5 

463684 

87. 

6 

1     41GG6|f/rt//o«s. 

15 

$812.25 

87. 

7 

57979fl| 

16 

$147,9375 

87. 

8 

5552-1 

17 

£14  145. 

87. 

9 

Imo.  \ivk.  A^da. 

18 

£166  2s.  Qd. 

87. 

10 

12  years. 

19 

6d. 

87. 

11 

^mo.  Ow.  5d.  14/i.  40m. 

20 

$6,95175 

87. 

12 

765  barrels. 

21 

$8,64 

87. 

13 

$72 

22 

$93. 

87.1 

14 

$0- 

88. 

23  if 

?7,875                   2 

8  SI  9 

33 

7680 

88. 

24  1 

8  cents.                2 

9G780C.  ft 

34 

1/6.  7oz.  12pwt.  Ugr 

88. 

25 :: 

6                           3 

0  $773, 39£ 

)  -io 

SIO 

88. 

2G';1 

olb.  6or.  lipivf.  3 

1  $4.2408 

36  2/>M.  \pk.  Iqt. 

88. 

27'$50                         3 

2  $16,702^ 

)  37  $0,75 

89. 

38 

104  sheep. 

44 

$59b2Sl 

89. 

39 

12  days 

45 

31680  times. 

89. 

40 

16  caunisters. 

46 

130  farms. 

89. 

41 

52gal.  Iqt. 

47 

n9/4¥3Ta«i-^s. 

89. 

12 

1424 

48 

S44397293 

89. 

43 

96  acres. 

49 

11/tr.  Ami.  32.SW.,  A.M. 

90.  601 

127°  '30 

57 

5''ll-/fl?.l  fur.oArd.2yd. 

90. 

51 

1 

J  67°  35'  jVs  Long. 

\  9hr.  I9in.  P.  M.,  B's  time. 

58 

1000000>. 

90. 

59 

\o82Arods. 

90. 

52 

lOcords,  ^  cord  ft.  \^c.Jt. 

GO 

36100 

90.  53 

\cwt..  '3qr.  9lb.  lOoz. 

61 

291111.  ofur.  2rd.  16/?. 

90.  54 

$164,475 

62 

1  Oacres. 

90.  55 

282?/r.  6mo.  8da 

63 

c.'oyards. 

90.'56 

6i:al.  2(jt.  Gpt.  2r/i. 

91. 

G4[. 

3//c/.  Iqr.  oiia. 

69  2o}ir.  6mo.  IGc/a.  'Jhr. 

91. 

6oi 

33  of  each. 

70 

$10591021,60 

91. 

r,6 

13209  + 

71 

)  $2478  widow's  share 
)  $1239  each  child's  " 

91. 

67  $11,88 

91. 

1 G  8 '  1  ?/  ;•.  2  0  5^/a .  1  Ih  r.  1 5  m. 

72 

$9 

92. 

73 

130C8  shuigies.  1  . 

J  107°  47'  diti:  iu  long. 

)   Ihf     1  i  «j.   ftsvr    (lifT.  in  timfi. 

92. 

1 ! 

( 

■ 

"     

414: 


ANSWEK5. 


P. 

EX. 

75 

ANS. 

KX. 

78 

AXS. 

92. 

1/i;-.  11?«.  8sec.,P.  M. 

$2 

92. 

7G 

j  Ahr.  5(J7n.,  P.  M. 

79 

4333|-  schoolhouses. 

92. 

\  2^°  from  N.  York. 

SO 

AQ^lbs. 

92. 

77 

48  hours. 

81 

14  days. 

93. 

82 

28bar.  6gaL 

I  4S2A«.  \pk.  -I'jis.          —  l:,t. 

93. 

S3 

24:bar.  ]9gals. 

90 

]  U\{)hu.opk.Qqt.  \^j)t.  —  2A. 

93. 

84 

$85,33i 

I  o2lbu.  2pk.  \qL  ^pt.  =zod. 

93. 

85 

1 1  2.     rolls. 

91 

I  40°  50'  East. 

93. 

86 

lini.  6/ur.  20rd. 

I  35^  days. 

93. 

87 

8750  pounds. 

f  $2400  =  Captain's  share. 

93. 

88 

$18,025 

(-^  j  $2000  =  2  Lieut:s      " 
•^"^   ]  $3600  =  6  Midship.    " 

93. 

89 

2500  barrels. 

93. 

[$  200  :nr  each  sailor  s  " 

94. 

93 

87°  30' 

98 

514  eagles. 

94. 

94 

9hr.  33m.  146rc.,  A.  M 

.       99 

2011  bushels. 

94. 

95 

lOhr.  54.m.  IGsec.  A.  J 

n  100 

$7410 

94. 

96 

19° 

101 

Ij/r.  338da.  22hr. 

94. 

97 

4800//f^s. 

— 

96. 

1 

3x3;  2x5;  2x2x3;  2x7;  2x2x2x2; 

96. 

3x3x2;  2x2x2x3;  3x3x3;  2x2x7. 

96. 

2 

2x3x5;  2x11;  2x2x2x2x2;  3x3x2x2; 

96. 

2x19;  2x2x2x5;  3x3x5;  7x7; 

96. 

3 

2x5x5;  2x2x2x7;  2x29;  2x2x3x5- 

96. 

2x2x2x2x2x2;  2x3x11;  2x2x17; 

96. 

2x5x7;  2x2x2x3x3. 

96. 

4 

2x2x19;  2x3x13;  2x2x2x2x5;  2x41; 

96. 

2x2x3x7;  2x43;  2x2x2X11;  2x3x3x5. 

97. 

6 

2x2x23;  2x47;  2x2x2x2x2x3;  2x7x7; 

97. 

3x3x11;  2x2x5x5;  2x3x17;  2x2x2x13. 

97. 

6 

3x5x7;  2x53;  2x2x3x3x3;  2x5x11; 

97. 

5x23;  2x2x29;  2x2x2x3x5;  5x5x5. 

97. 

7 

2x151;  5x01  ;  2x2x151;  5x5x5x7; 

97. 

3x5x5x13  ;  5x131. 

97. 

8 

5  X  3  X  2. 

97. 

9 

2  X  3  X  7. 

97. 

10 

3x5x7. 

97. 

11 

2x3x7. 

97. 

12 

2 

97. 

13 

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lUO 

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ANSWEKS. 


415 


P. 

EX. 

101. 

1 

101. 

2 

101. 

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101. 

4 

101. 

5 

ANS. 


16 

7 

22 

124 
62 


EX. 


6 
7 
8 
9 
10 


ANS. 


81 

45  bushels. 
25  acres. 
12  feet. 
3  bushels. 


EX. 


11 


ANS. 


$22  jyer  head. 
13,  A  bought. 
21,  B       " 

29,  a     " 


103. 

1 

1260 

5 

10500 

9 

103. 

2 

7200 

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10 

103. 

3 

12G0 

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540 

11 

103. 

4 

1008 

8 

420 

12 

336 
1176 
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$1680 


12 


112  men  at  $15 

105   "  $16 

80   "  $21 

70   "  $24 


104. 

r  210  b 

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1 

60  days. 

104. 

bags     105    times. 

A,     3   times. 

104.' 

13 

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■< 

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104. 

boxes     30       " 

C,     5       " 

104.! 

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7 

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12 

36  pounds. 

106. 

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14 

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106. 

4 

48 

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106. 

5 

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1 

107. 

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107. 

16 

107. 

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IS 
19 
20 


15  barrels, 
6210  bushels. 
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21 


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115. 

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3 

33.331    13 

<^  2505.068  .  - 
25050.68 

4162.2 

165. 

4 

1.0001 

1  41622. 

16.5. 

5 

12420.5 

250506.8 

416220. 

165. 

6 

.005 

4162200. 

165, 

7 

4.25 

r  48.65961 

(  254.7347748 

16.5. 

8 

.007 

4865.961 

25473.47748 

165. 

9 

.075     14 

■{   48659.01   ^ 
486596.1 

254734.7748 

165. 

10 

1.27 

]  2547347.748 

165. 

11 

.015 

4865961. 

25473477.48 

165. 

12 

17.008 

1 

254734774.8 

ANSWERS. 


423 


p. 

EX. 

165. 

17 

165. 

18 

165. 

19 

165. 

;^o 

ANS. 


.13956463  + 
1918.515  + 
.U0473O 
174.412+ 


KX. 


21 

90 


24 


ANS. 


69.7125 

1.36S32  + 
12976.81  + 
.004958  + 


EX. 


25 
26 

27 
28 


ANS. 


6.165c.  yd. 

$9,875 

$2.15 

$0.62 


166. 
166. 
i6G. 
166. 

167. 
167. 
167. 


29 
30 
31 
32 


18  2)oujids. 
8  suits. 
14  days. 
55.5  bush. 


269  acres.  $13573.204  cost. 
$50,458  +  ,    average  jvice per  acre. 
$7631.8855,  elder  s  share. 
$5723.914125,  each  of  others. 


10970 
G0200 
1000 


100 

10  ;  100;   1000  ;  30;  20;  2000;  12; 

1200;  500000. 


168. 


170. 
170. 

170. 
170. 
170. 
170. 
170. 


,375 


8.311+  II  4    1.563  + 

4 
5 
6 

7 
8 
9 
0 


1.1604+    I  6     16.119  + 


79.1188 
35.2843 
11.5834036 

;5i3202.8S69 


.25  ;  .5  ;  .75 
.8;  .875;  .3125 


015625;  .2666  + 
125;  .003 
2571+  ;  .4411  + 
23903  + 
07157  + 
4375;  .078125 
00448 


11 
12 

13 

14 
15 
10 
17 


,536;  .372 
.9 
,7333  + 

,48375 
,5128 

,5375;. 0056  + 
,1666-f 


171. 

18 

171. 

19 

171. 

20 

171. 

21 

171. 

22 

1.000  + 

.15909  + 
$100.80 
$17.R5 
30.011  + 


24 
1 
2 

o 
O 


2.9166  + 
2.8412 

JL.  3 

4  '  4 

8  '  8 
2  1.    1 

4 

5 

6 

7 

8 

16  03 
2  0"0  0 
.=  4  7 
800 

3 

1  fiO 

903 

4000 

200  '  400 

R4 

jiOJl 
5000 


172. 
172. 
172. 

172. 
172. 
172. 


.0546875/6. 
£.325. 

.olQda. 
71.1511  +  wz, 

.6625/6. 


/ 

8 

9 

10 

11 

12 


.15375  ions. 
£.1225 
.26175^. 
.100511+ m?, 
.&Acivt. 
.91111+  lb. 


1  '^ 

14 

15 

16 

17 

.875yd. 

.01587+  hhd. 
.7129975c/a 
.2325  ions. 
£.9729  + 


173.' 

18 

173. 

19 

173. 

20 

173.; 

21 

173. 

22 

173.: 

23 

173 

24 

.48125.4. 
.5'oE.E. 

.0016177  w^^. 
.25625^ 
.0041956  7: 
.10416  cAa/. 
.00994318m/, 


25 
26 
27 
28 
29 
30 
31 


.791666  yr. 

.9111/6. 

.3375 

.3125  chal. 

.0409»ii. 

.01875  ream. 

.02026rd 


00 
34 

35 
36 


.19672y?-. 
.3489  fe 
.01537  +  /iArf. 
.005A 
.ol25yd. 
3.390625/i!. 


174. 
174. 


1  J   2qr.  nib.  4oz. 

2  I   Ih/id.  V2,gal.  oAAqt. 


o 

4 


16.s\  Id.  2.99/ar, 
2gal.  \qt. 


424: 


ANSWERS. 


P. 

EX. 

5 

ANS. 

EX. 

ANS. 

174. 

(  Iwk.Ada.  2ohr.  59m 
\  56.0SCC. 

14 

(  o2mi.  \fiu:  Urd. 
I  Uft.  9.4  0S/«. 

174. 

174. 

6 

b.F. 

15 

2/1.  1.5in. 

174. 

7 

6cirL  3qr. 

16 

4  5    13    19   9.(Sgr. 

174. 

8 

l/jhd.  Algal.  \qt. 

17 

'6R.  \P.  \3.3 \sq.1jd. 

174. 

9 

20!/al.  1<7('.+ 

18 

9  sheets. 

174. 

10 

lOos.  ]8pwt.l5.99gr. 

19 

nibs. 

174. 

11 

oqr.  I. Qua. 

20 

Id.  2/ur. 

174. 

12 

5/1.  11.9  4-  ill. 

21 

IE.  UP. 

174. 

13 

j  24:F.  23i!q.yd.  5sq.ft. 

22 

286da.  nhr.  18'  36''' 

174. 

\        82.4832sg.  in. 

176. 

1 

.06 

5 

.029729  + 

176. 

2 

.09285  + 

6 

.034 

176. 

o 
O 

.034370 

7 

.028 

176. 

4 

.013281  + 

8 

.043055  + 

179.1 


2.  .    _fi_ 
-■}  '     3  7 


1  0    . 


15.  •       1_ 
3  7'     11 


4 


143 


_4_  •     1 
1  1    '     7 


180. 
180. 


4 
5 


J}.  • 

3fi   ' 
34   . 


269 


29 


_v.    .^.  ..«  .  •'^74 

'  4"  9  "5  '    "BTTe  '    "^ '  9 


2  1  7   .       1_  .     A  1 2A6 
I      45'     495'      75'         16fi5 


9    .     223   .      7  54  31 
0'     330'     99  9  99* 

.         ]  6  3       .      4J. 

'      1  fi  5  0  0   '     9  0 


182.11   2  |.]875'.||3|.0'0344827  +  ||4P09756'  ;    .'592';. 5^3 


2.4M8181S' 
.5M)25025' 
.008^497133' 


165.  urn  61 U/ 
.0P0I0  104' 
.03"777777' 


;; '  •  >  Q  o  .1  o  .->  ^ 
.0  00 O.J 00 

.4'757o75' 
J.7'577o77' 


9o.2'829647' 

09.74^203112' 

55.6^209780437503' 


0 
6 


47.3'763']90' 
216.2^542870' 


185. 

2 

457^757' 

6 

185. 

3 

2.9^957' 

7 

185. 

4 

5.09 

8 

185. 

5 

.6o'370016280907' 

9 

186.1 

2 

186.! 

3 

186. 

4 

186. 

5 

186. 

6 

186. 

7 

186. 

8 

186. 

9 

5.53780^5   , 

1.093^)86' 

1.6411^7 

1.7183\39' 

1.1710^037' 

G.rOSG' 

ll.V)(;8735102' 

.81G54U68350' 


4.37^4 
4.619^525' 
1.0923^7 
1.3!  62' 937' 


o 

4 
5 

6 
7 
8 
9 


13.570413'96103S' 

35.024 

7.719'54' 

26.7837M28571' 

3.r45' 

3.'82a5294 117647058 

1.2' 6 

15.43'423' 


ANSWERS. 


425 


loi. 

193. 
193. 
193.1 
193.1 


38 


2     56 


12 


4     40 


96 


o 

4 
0 


6 
7 

8 


i 

6 
2 
3 
1 
3 


9 

•10 

11 


5 

■A- 

9 
TT 


195. 

195 

195. 

]95. 

195. 

195. 

195. 

19.5. 

195. 


1 
2 
3 

4 
5 

6 

7 
8 
9 


$330 

$90 

504  miles. 

$2,08 

$875 

99  pminds. 

$2762,50 

$20 

$122,85 


10 

*> 

11 

12 

13 

14 

15 

16 

17 

18 

1400  pounds. 
16485  miles. 
$121,871. 
216  shillings. 

$3533,936-}" 
$86.62 
£39679.105. 
$39,371 


19 
20 
21 
22 
23 
24 
25 
26 


bo 


§382 

S63 

$0,036 

$2,52 

$1,925 

$2,10 

$52,50 


198. 
196. 
196. 
196. 


27 

28 
29 
30 


$7200 
137,909  + 
$132,589  + 


31 
32 


$18,66f 

35 

$56,355* 

36 

106|-  yards. 

37 

40  weeks. 

J 

$112,86 
$5427 
2l^a^.  wafer. 


197. 

38 

197. 

39 

197. 

40 

197. 

41 

197. 

42 

A,   $2142; 
$0,62|. 
G|-  bottles. 
1261-1  shilli 
168  povnds. 


B,  $1125, 


ngs. 


43 
44 

45 
46 
47 


9o 
55 

$1 
6/i 
14 


|-  gallons. 
2  miles. 
7444. 
o 

0// 


'6Zm.  A^^jsec, 


19 


i26 


ANSWERS. 


P. 


197. 
197. 


198.i 
198.1 

198J 

202.1 
202.i 
202.1 


EX. 


48 
49 


ANS. 


A,  155  miles;  B,  124  miles. 


14.  days. 


EX. 


50 


ANS. 


22^  days. 


51 
52 
53 


As,  $88,40  ;  £"s,  $77.35. 
10/i?-.  40m.  36^yec. 

48??i.  l^sec. 


54 
55 


$24,66^ 


3* 

16^  times. 


1 
2 

o 
O 


9  yards. 
8-|  roc/^f. 
]  CO  yards. 


4 
5 


7-1-  (iays. 
10 


6 

7 


920 

54  days. 


203. 

203. 
203. 
203. 


8  225  f%5. 

9  13  ounces. 
•  j  588000/6. 

I  546000Z/^. 


10 


1] 

12 
13 


{ 


588000Z6. 
14  ounces. 
20  f/ays. 
54  f/ays. 


14  12  days. 
15'   6     " 

le'ieo  " 

17:40.47. 


18  120  mn. 

19  51  da;/s. 


204. 
204. 
204. 
204. 


20 
21 
22 
23 


45  7nejt. 
13f  ounces. 

13 jY  (^ia!/^- 

201  days. 


24  11  Of  rods. 

25  ho'^days. 

26  8j-L  cioi. 

27  136  me?z. 


28 
29 
30 


6  horses. 

857 1|^  plan/cs. 

3  hours. 


207. 
207. 

207.' 
20J^ 

208. 
208. 
208 

210. 
210. 
210. 
210. 
210. 
210. 
210. 
210. 
210. 


1 
2 

o 
O 

4 


161  dai/s. 

5 

7200  men. 

6 

\%l^mihs. 

7 

72  acres. 

8 

10  daus. 
d2\  days. 
$36. 
292.5  ^aZ. 


9 

10 

u 


1 156  tailors. 
19600  men. 
50  wie;*?.. 


12 
13 
14 


$471,04. 

■3\^da. 

180. 


15j600. 
16  14f  fZa. 
17|7i-oz. 


18 
19 
20 


971/6. 
32  days. 
32  horses. 


215  vien. 
22|132  days. 


$1000,  A's. 

$1200,  i;'5. 

$  800,  C's. 
$1714,28A  ^V 
$285,7 If,  B's. 
£4030,  ^'s. 
£3980,  //.v. 
£3980,  C's. 
£4010,  i>'s.  , 


'  f=5000  As. 
§2500  i?V 

<^  .$33331  CV 

§2500",  D's. 
$6666f,^'*'. 
100,  As. 
140.  B's. 
200,  (7'5. 

OClOO-.r,Jl  .S. 


$3000,  B's. 
$3000.  C"5. 
$26G6f ,  Z)'s. 
$1500,soM'.'?s/t. 

$3000,?//o'.9s/i. 
$12961,50  .fs. 
$1 5737,25  ^'s. 
$10802,25  (7'5. 
^l^ooj/sgiiin 


211. 

211. 

9 

211. 

211. 

211. 

10 

211. 

211. 
211. 

11 

$450. 

$600. 

$750. 

As,  $4242,50  s 

7i'.v,  15939,50 

C".s§.6788 

$237,75,  As. 

$181.0025,  B's 


tack :  $;1697  gain. 
"  §2375,80     " 
"  $2715,20     " 


11 
12 

13 


($125.4375  6' 
I  $70,       .D's. 

$12720. 
f  $87,831 +.4 

$65.06 -f-  B. 

$18.795+C 

$68,313  +  Z) 


ANSWERS. 


427 


p.  1 

EX. 

ANS. 

211.! 

(Sl015,33i 

the  first. 

211. 

14 

■}  $1523,00 

"    second. 

211. 

'■    (S2030,66f, 

"    third. 

212. 

i  §16,38,  A's. 

r> 

($6577.23A±3^^'5. 

212. 

1 

{  $35,10,  £'s. 

|$1S22,76L46^^'5, 

212. 

1  SI 8,72,  C's. 

(  $288,  A's. 

212. 

2 

S7. 

4 

\  $270,  B's. 

212. 

— 

I  $2-10,  C's. 

213. 

5 

J  $280,  D's. 
}  $163,  C's. 

r  $84,  A's. 

213. 

8 

$90,  B's. 

213. 

($1309,43^;^^,  oj/i 

:ers. 

]  $82,50,  C's. 

213. 

6 

■\  $2946,22f|f,  midshipmen. 

$90,  D's. 

213. 

(  $10504,33^1^,  sailors. 

($2,  \  St  grade. 

213. 

1  $2648,86-i-\.  A's. 

9 

■^  $  1,  2d      " 

213. 

7 

n2901,13-rV,  ^'5. 

(  $0,50,  M   " 

213. 

(  $1850,            CVv. 

10 

\  $800,  5'a'  stock. 

213. 

\  1  5  Hio.9.  C's  time. 

214. 

1 

.095;  . 

0575. 

3 

2.08, 

3.75 

;  .95. 

214. 

2 

.125  ;  .09875. 

4 

.666f. 

215. 

2 

$3,14 

9 

16.74  nid.es. 

16  $4344,35 

215. 

3 

$4,7825 

10 

47.725  sheep. 

17l2625/;ar. 

215. 

4 

■i. 562  5r/ds. 

11 

27.54  tons. 

18'$5144.625 

215. 

5 

2.S3937  ociot. 

12 

$300,365. 

19  $12500 

215. 

6 

\.002lbs. 

13 

15,75  coius. 

20  $3867.01875 

215. 

7 

126?/. 

14 

1  60  bales. 

2l'$15000 

215. 

8 

$90. 

15 

478.1 25yi?. 

22  $65 

216. 

23 

742,85  gallons. 

26 

$6093,75. 

216. 

24 

205  boxes. 

27 

$196,59375. 

216. 

25 

.421;  $10625. 

"   i 

217. 

1 

.20 

5 

.25 

9 

.01375 

13 

.05 

217. 

2 

.125 

6 

.875 

10 

.33^ 

14 

.035 

217. 

3 

.075 

7 

.625 

11 

.375 

15 

.72 

217. 

4 

.136 

8 

.0075 

12 

.125 

218. 

1   1      %'i  per  head.  \    2     $5425    |3     $50000      4     $5000 

220. 

2 

$60.9875 

7 

$283,8438 

12 

$373,2495 

220. 

3 

$221,91 

8 

$422,8976 

13 

$735 

220. 

4 

$360,2832 

9 

$1112,90 

14 

$1016,075 

220. 

5 

$473,844 

10 

$265,2345 

15 

$120.80 

220. 

6 

$1312,5 

D 

1 

1 

$1893,75 

10 

$5796 

423 


ANSWERS. 


221 
221, 
221 


EX. 


2 
3 


ANS. 


$20,9U9 
$26,313 

$458,88 


EX. 


4 
5 
6 


ANS. 


$1979,5013 

$5618,75 

$628,4162- 


EX. 


7 
8 


ANS. 


$64,0625 
$157,65625 


o 

4 
5 


$42,2432  + 
$420,2531-1- 
$213 
$181,25 


$11,0415 
$132,7707-1- 
$26,95864- 
$416,16734- 


10 
11 
12 


1334,21874- 

$120,0694- 

$40,0908 


13 

14 

15 

16 

17 

18 

*">  1 

ol 


33 


2 

o 
O 


11 

12 


2 
3 


2 

o 
O 

4 


$81,67784- 

$162 

$221,266 

$389,24  66 

$135,3714 

$42,94044- 


$933,1574- 
$499,3394- 
$140,6444- 


19,$84,6855 

20855,66854- 

2l|$32,6664- 

22:$8590,8324- 

23l$36 

24!$93,78434- 


25  i  $160,44084- 


26 
27 
28 
29 
30 


$12,9644- 

$82,0364- 

$70,964 

$879,467 

$801,769 


34 
35 

36 


$5085 

$403,858 

$9337,50 


1 

2 


$394.325-f 

$697,986 


226. 
226. 

3 
4 

$3339,613 
$823,9024- 

5 
6 

$4640,5324- 
$1976,634- 

227. 

227. 
227. 

2 

o 
O 

4 

£45  8s.  l^d. 
£45  12.S.  4  It/. 
£154  7s.  Od.  2  far. 

5 
6 

1    7 

£1133  106'.  9}d.+ 
£199  6s.  3^d. 
£6  16s.  5cl. 

$3976,7824- 
$439,80 
$6234,76-h 
$30000 


5 
6 
7 


$952,576-(- 

.07 

.09 


8 

9 

10 


.10 

.05} 

.121 


2yr.  67710. 
omo.  18da. 


13 

14 


lyr.  Amo. 
i6yr.  8mo. 


15 

16 


Syr.  4??io. 
lyr.  6mo.  20fl?. 


$5359,3664- 
$8925.544  4- 


4 
5 


^1127,041 
S190,758 


G   I   $156,204- 


$25,3575 

5j291,7215 
$57,3048 


5 
6 


$73,015 
$83,20 


7 

8 


$845,837 
$48165,936 


234. 
234. 

9 
10 

$14523,555 
$926,744 

11 
12 

$8501984,90622" 
$124,1624 

235.    13 

$151,5811         14      $16,3875    |    15      $445,857 

236. 

1      $562.50            2  1  $184,499  4-        I    3      $21 

\ 


ANSWERS. 


42\^ 


p. 

EX. 

4 

ANS. 

EX. 
10 

ANS. 

236. 

$5000 

($3538, 083  CMA'Ava/. 
($388,083  gain. 

236.1 

5 

$1902,557  + 

236.i 

6 

($2763,703  2^r.  vol; 
\  $236,297  disc'L 

11 

$9890,239 

238. 

12 

$10,890  loss. 

236. 

7 

$4820,537 

13 

.00414,  a^ic/s. 

236. 

8 

$4800 

14 

236. 

9 

$1379,6123  + 

15 

($2369,2617c«sAmi 
($61,9883  diff. 

236. 

— 

240. 

1 

^6,15 

5 

84,374 

240. 

2 

$7,65 

6 

882,591+  gain. 

240. 

o 

j     2'i,2'^\^  discount. 
\  $476,708^    pres.  val. 

7 

$11,785  difference. 

240. 

O 

8 

$15,4044  diferejice. 

240. 

4 

81 225,3555  ^9/-e6\  val. 

9 

8981,21   cash  value. 

241. 

2 

$296,50               1   3      $697,20              4      $474,375 

242. 

5 

$3522,092                      

243.; 

3 

$34,8375 

H 

$25420,195 

243. 

4 

$164,53125 

9 

86835,283 

243. 

5 

$96,33     $5831,67 

10 

8935 

243.! 

6 

(  $163,80  commission. 
\  $4340,70  whole  cost. 

11 

\  863,625  cotn'n. 

243.! 

I   4544.642+ 6 wi-Ae/s. 

243.' 

7 

8115,39  + 

244. 

12 

$255»,16 

15 

15   tons. 

244. 

13 

(158  barrels. 
\  $2412.66 

16 

870 

244. 

17 

20  shares. 

244. 

14 

$420,922 

18 

$55743,289 

216. 

1 

85320 

1     3 

859110 

216. 

2 

$666 

4 

821375 

247. 

5 

$7999,6875 

7 

$300 

247. 

6 

$213500 

1 

$3529,41  + 

248. 

2 

56  shares. 

5 

$6000 

248. 

3 

$4000 

6 

$10432,432  + 

248. 

4 

$7235,142  + 

249.  1     1 

.08 

o 

.08 

5 

.4166  + 

'249. 

i      2 

.20 

4 

.03 

6 

.05 

250. 

250. 

2 

o 

7  per  cent,  the  best. 

a       i         41,^       h^r.t 

4 

$166,66| 

o    1     o  pvr  i;cnc.  lug  (yco(.          }j 

251.11   1   1   $33,75 


2  1  $0,56 


3  I  $236,25 


430 


ANSWEKS. 


P. 

EX. 

ANS. 

EX. 

ANS. 

EX. 

4 

AXS. 

252. 
252. 

1 

2 

$170 
548,80 

$6,0053 

$70  gain. 

253. 
253. 
253. 

1 

2 
3 

$0,90   ) 

$3,20 

$0,96 

4 
5 
6 

$5,70 
■ii;  18,03 
$0,66 

7 

$1,80 

254. 
254. 
254. 
254. 
254. 


2 

o 
O 

4 
5 


.18 

.25 

the  same- 

.80 

.25 


7 
8 


$160,34375   wliole  gain. 
]  .046+  2->e?*  cent. 
.45 
$25,65  lost. 


263. 

265. 
265. 
265. 


1 
o 


255. 
255. 

9 
10 

.U5  ^;er  cent  loss. 
.40 

11 
12 

$508,50 

256. 
256. 
256. 

1 
2 

o 
O 

§50  168,59 
S158,40;  8237,60 
$126;  $252 

4 
5 
6 

$300 

$89,55 

$47,8125 

7 

8 

$1252,125 
$163,80 

257. 
257. 
257. 

9 

10 
11 

$16481,25  loss. 

.051 

.Olf 

12 

13 
14 

.04i 

$2i'ooo 

$9020 

15 
16 

$127,4625 
$298,2546 

259. 
259. 
259. 

1 
2 

o 
O 

$121,72 
$232,50 
$262,50 

4 
5 
6 

$20 

$98,20 
$120 

7 

$9101,635 

260. 
260. 

1 

2 

$411,15 

$757,908 

O 

4 

$1227,395 
$1318,94 

262. 
262. 
262. 

1 
2 
3 

$7051.63415 
$9049,53795 
$23058,6765 

4 
5 

$16355,52 
$2160,90 

$2159,6134- 
'g^per  cent. 
$37901125 


1^  per  cent. 

$82,25 

$56,9075 


266. 

4 

^perct;   $15,50 

(.015  on  $1 
\  $112,50 
($18 

266. 

5 

$5820 

8 

266. 

6 

$22236,197 

266. 

(  $4656,05  whole  tax. 

9 

($7,40 
■[$9,225 

286. 

7 

\  .005  on  $1  ;  $27 

266. 

(  $6,8775  6^'s;$12,78ir'.v. 

269.11  3  I     $260,9932 


4  I       $713,37 


ANSWEKS. 


431 


1-. 

EX. 

ANS. 

9 

ANS. 

270. 

5 

iVr.  l^rwt.lqr.  16.G8/6. 

$1190,343  + 

270.! 

6 

3  T.  Inot. 

10 

$744,546 

270.i 

7 

\  GT.]3cwt.2gr.  Alb. 

11 

$250,835 

270.; 

\  $308,4774 

12 

$4,09  + 

270.' 

8 

$792,612 

13 

$125,S0| 

271 

271, 

271, 

271, 

271 

271. 

272. 


14 

-S39S.il  99  + 

20 

$1512 

15 

.$'166,273 

21 

$423,36 

. 

16 

$1101,24;  $0,14 

22 

1251,453  + 

17 

$7936,50 

23 

$1457,75 

. 

18 

$820,4625 

24 

(  22.605cdi';.  tare. 
($68,5856  duty. 

. 

19 

§16,206  + 

12  months. 


273. 
273. 
273. 
273. 

2 
3 

4 
5 

9  mo. 

B^mo. 

1  mo.  oda. 

64-mo. 

4 

6 
7 
8 

21+  daj/s. 

6mo.  Gda. 

2(j^da.,  or  on  July  23 

274. 
274. 

9,^1 

io!s 

S^'^Jjda.  or  on  Sept.  19 
ISl^da.  or  Dec.  30th 

11 
12 

7Sfy(/.,  or  Oct.  19 
59.(57da.,  or  June  28. 

275.1!     li*0,50^ 


276. 
276, 
276, 


2 

o 
O 

4 


80,66 
$0,49 
$1,00 


o 
6 
7 


75" 


19  carats. 
$0,13i 


80,30 


278. 

278. 
278. 


1 
2 


{1 


lb.  at  Sets.  1  lb.  at 
Qcts.  3 lb,  at  lActs. 
Mb.  each. 


1  calf",  2  cows,  1  ox,  1  colt. 
3  gallons  of  water. 


279.11    I    I   20   pounds  of  each. 


2  I   75  pounds  of  each. 


280. 
280. 
280. 


oG(/al.  at  7s.,  24/7a/.  at  7*'.  6c/.  and  at  9s.  6c/.,  I2gal.  at  9s. 

10  at  $2,  15  at"'Sf. 

2olb.  at  5  and  7,  100  at  7Jc;s.,  37^  at  9^,  and  50  at  \Qcts. 


281.! 

1 

22  pound  of  each. 

281.1 

2 

9gal.  of  water,  40^  at  $2,50,  13^  at  $3. 

281. 

3 

12  calves,  12  sheep,  16  lambs. 

281. 

4 

8  at  $6,  8  at  $7,  4  at  819. 

281. 

5 

^Qfjal.  at  4s.  and  \Qgal.  each  at  6?.  8s.  and  10s. 

281. 

6 

6  vests,  12  pants,  6  coats. 

281. 

7 

30  at  15  carats,  and  4  each  of  20c.,  22c.,  24c. 

281. 

8 

10  at  81,   15  at  81,  10  at  85. 

288. 

1 

88591,975. 

432 


ANSWERS. 


P. 

EX. 

AN  3. 

EX. 

ANS. 

289. 

2 

$8637,168  + 

2 

$176204,4729 

289. 

o 

S9777,636 

0 
0 

£14014  17s.  7J.+ 

290. 

4 

$6005,368 

2       .07  'p<:r  cent  al/ovepar. 

290. 

5 

$807,874  + 

291. 

3^^12286,06 

0 

(§1250,52 

291. 

4184097  f'7tcs66 centimes. 

( .03  ijer  cent  nearly,  below  par. 

291.':li$0657,G93 

- 

299. 

1  1 

22Dj\  ions. 

3|729^V5^««^'- 

5   1006,57  7. 

OQO 

i  o 

43841  ^^"*'- 

4  300.14  r. 

i 

^yy.  1  ^ 

301. 

116 

15  iH                      1 

29,76,765625 

301. 

2225 

16 

1225 
7056 

30!  10,4  976 

301. 

3:676 

17 

1  "  fi  2  h 
TT 1  f )  0  9 

3134012,224 

331. 

4 

20164 

18 

7,84 

32.0184528125 

301. 

5 

214369 

19 

58,140625 

'^'^14  09  6 

301. 

6 

1795600 

20 

250AV 

34iM 

301. 

7 

605,16 

21 

51030,81 

3  ^      8  1 

301. 

8 

.276676 

22 

216 

36  3l54|f 

301. 

9 

9,765625 

23 

13R24 

-^^  ^^'10  2  4" 

301. 

10 

.00274576 

24 

373248 

3R 390H2  5 
"^153  1  44  I 

301. 

11 

60639,0625 

25 

1953125 

39  i48a6,936 

301. 

12 

9 

26 

2515456 

40  ,( 

)002441 40625 

301. 

1  •  >  3  6 

27 

20736 

41  2 

893640,625 

301. 

1449 

28 

59049 

12  1 

0,272025 

307. 

1 

7 

8  2 

.5 

15 

.1581  + 

22 

3 

7 

307. 

2 

12 

9  1 

6,7 

16 

.779  + 

23 

.2828  + 

307. 

3 

15 

102 

505 

17 

.149 

24 

11.618  + 

307. 

4 

48 

118 

9,409  + 

18 

5,01 

25 

137,84 

307. 

5 

6 
9 

12.^ 

153 

19 

14,015 

26 

.885  + 

307. 

6 

15 
4  0 

13.5 

33  + 

20  1,2247  + 

27 

75.15 

307. 

7 

.1 

4 

14.i 

)68 

24 

21 

53 

28 

400.06 

309. 

1 

30//. 

2 

221  stones. 

309.1 

1 

343 

3 

21y9j-  rods. 

310. 

4 

(  GOrds.  wide. 
I  \SOrds.  lonq. 

8 

94,708/7. 

310. 

9 

53.331/1!. 

310. 

5 

10J.0/^.29P.168f^5'.//. 

10 

8,660 //!.+ 

310.! 

6 

15  ft. 

11 

825,8  mikfi. 

310.! 

7 

1 

35/V. 

12 

$100 

ANSWERS. 


433 


r. 

EX. 

AiNS. 

EX. 

ANS, 

311. 
311. 
311. 
311. 

13 
14 
15 
16 

loft. 

28.28 +/<. 

6i/(. 

11.041rf/5. 

17 

(  4.405  + i/i.,     1st  man's  share. 
-'  5.739 +  ni.,     2d      " 
(  13.856+?;;.,  3d      " 

315. 

1 

12 

5 

179 

1 

2,028  + 

5 

.729  + 

315. 

2 

49 

6 

364 

2 

12,0016  + 

6 

.0]5 

315. 

o 

36 

7 

439 

O 

.232  + 

7 

.188  + 

315. 

4 

247 

8 

3072 

4 

27,0002  + 

8 

4,339  + 

316. 

1 

316. 

2 

316. 

3 

316. 

4 

316. 

0 

316. 

6 

316. 

7 

316. 

8 

316. 

9 

316. 

10 

5 

9 

4-1- 

7. 
B 
9 
2.S 
2  7 
f>4 

35 

1.987  + 
3.83  + 


1 
2 

3 

4 

5 

6 

7 


{ 


27/V. 

19//;.,  length  of  each  side. 

2l66s5'./(f.,  area 

36/?.,  length  of  each  side* 

8.57+//. 

9.77 />.,  length  and  breadth. 

19.54 +/;;.,  height. 

10.125  c?i. /it. 

45  cents  per  yard. 

2025  whole  number  of  yards. 


317. 

9 

317. 

10 

317. 

11 

317. 

12 

317. 

13 

317. 

14 

317. 

15 

317. 

1 

317. 

16 

317- 

319. 


64/65. 

8//.,  length  of  each  side. 

8  jrlobes. 

$1331 

12m.  long,  6m.  wide,  \m.  thick. 

24/.  long,  20/.  wide,  9//.  deep. 

20  feet. 

.54  + Mi.,     1st  woman's  share. 

.69  +  ^;.,     2d         " 

.99  in.,       3d         "  "        4th,  3.77m. 

'89  \     2    1     $80  II    3    I     §396 


320. 
320. 

321. 
321, 
321, 


4 
5 


1 
2 


174i'(/A-. 

20 !/.,  to  bring  back  the  nearest. 


1 
2 


5  miles. 

$2 


fftw. 

ij;2730 

§64,96 


79l-l-m2. 


I0??z«.,  7/Mr.,  27rds.,  \\yd. 


322. 

i    1 

5551^*2/.        1    2    1    13rfff.;  'M2ini.   |    3   |   6 

324. 

i    2 

3125000 

o 

4^ 

325. 
325. 

4 

5 

100000000000000 
.$3200 

6 

7 

$54000 

$327,68 

434 


AKSWEK3. 


P. 

EX. 

ANS. 

EX. 

4 
5 

ANS. 

326.1 
326.' 
326. 

1 

3 

118081 

2044 

811184810 

$4294  9672,95 
93S249922+  ships. 

34i.';2i 

$4166,40 

28 

2i|-  hours. 

32 

117/A. 

36  $5600 

341. 

;2o 

9f  days. 

29 

OO 

S693 

37 

$0,071 

341. 

26 

36  feet. 

30 

$100          1 

'34 

1 1  cents. 

38 

$126 

341. 

27 

§1770 

31 

5i  yards. 

j35 

9;56 

39 

$44,10 

342. 

-10 

$43 

r    60  the  first. 

342. 

41 

$240,75  gain. 

47 

100     "     second. 

342. 

42 

5//.r.  21  mi.  l&^jscc. 

1  140     "     third. 

342. 

\o 

40  yards. 

ISO     "     fourth. 

342. 

14 

10  hours. 

48 

16-i  inches. 

342. 

45 

36  days. 

49 

2y^;j  months. 

342. 

IC 

11^  days. 

50 

(  35J-  yards  baize. 

342. 

I  $2,20  per  yard. 

343.  51 

$12                                   1 

r-^^2317,15.4's. 

443. 

52 

$1,20                                ' 

'\<-. 

$1853,725'.?. 

443.  oo 

'S7  S.  652  +  discount. 

OO 

i 

1  $2317,15  6"6\ 

343.  54 

8129,60 

827S0,58Z>'s. 

343.  ..- 

\  $19,375   most  advan- 

f  $95,10.1'5. 

343. 
343. 

56 

]       tajTL'ous  for  cash. 
1292^.823  gain. 

59 

$95,l0i>"6-. 
'   $133,1 4  (?'s 

34:}. 

(SI  22,7  O^'^'. 
\  Si 63,60 Z^'s. 
(§1 96,32  6"6-. 

$l52,16/;'6'. 

343. 

57 

60 

71  ounces. 

343. 

G] 

8f  days. 

343. 

— 

1 
1 

;62 

17  times. 

344. 

63 

41|  days.                   II 

(  8172,78+  more  advanta- 

314. 

64 

5  months  24  day-s.    167 

<       geous  in  bond  and  inort- 

344. 

65 

68   days.                     i 

(       gaae. 

344. 

66 

126":allons. 

68 

$3312,417  + 

344. 

70 

1 

69 

§42,60 

1 

345. 

4  yards  high.  •                  [ 

77 

836000 

345. 

71 

20  hours  ;  140  miles. 

78 

41,183+  bushels. 

345. 

72 

f  S2  the  lirst. 

79 

137.942+  feet. 

345. 

(  $6   the  second. 

The  second.  10  dava 

345. 

73 

100  thousand  leet. 

ailer  the  od  ;   thv.' 

345. 

74 

91  39 
~  '  5  0 

80 

<       first,    8  days  after 

315. 

75 

$3825 

the    second,  or  I"' 

345. 

76 

i>144,  better  to  pay  casli. 

days  after  tlic  .'i  1. 

ANSWERS. 


435 


p. 


346. 
346. 
346. 
346. 
346. 
346. 
346. 
346. 
346. 


EX. 

ANS. 

EX. 

ANS. 

81 

§6890 

86 

12ibs.  soap. 

82 

(  $13 16,  whole  cost. 

87 

5  o'clock  20m.  P.M. 

(  -^7,  cost  per  acre. 

88 

$f  =  $0.66f. 

83 

32  days,  or  Mar.  1  6. 

89 

j  24  chickens. 

84 

(512     slabs; 

(36  turkies. 

"[  $302,222.,  cost. 

1  $350  A'&;  $297,50  B's; 

85 

\  $210   C's;  $175  D's  ; 
f  $122,50  E's. 

347. 

90 

347. 

\ 

347. 

91 

347. 

i 

347. 

347. 

92 

347. 

347. 

347. 

93 

8  days. 

$1707,50  first, 
$2157  second, 
$2516,50  third. 
$960  stock; 

$180  gain,  first. 
$640   stock; 

$120  gain,  2d. 
49.945+  leet. 


94 
95 

90 

97 
98 
99 


8 

llri/i;- 
(  $5331+  A's. 


oi'  a  week. 
1341 


miles. 


\  $8&8f  +  B's. 

(  $177,H  C's. 
36^  days 

84485,006     feet. 

$200,06  in  favor  of"  1st  invest. 


348. 

100: 

^3599680  cubic  yards. 

102 

$4004,338+   5th. 

348. 

101=! 

^4646,363 

1032160    men. 

348. 
348. 
348. 

102 

'$1555,017+    1st. 

$1354,717+    2J. 

'    $4304,663+    3d- 

104 
105 

j  $57,142  A's. 
I  $42,857  B's. 
$31 

348. 

1 

^$5781,263+   4th. 

106 

8  hours. 

(  $30  com.  diff 
i  $2160  whole  cost. 
'$144,03   A's 

$  90,12    B's 

$  63,45  C's 

$168,35  D's 

.06397 

$14467,505 


111 

112 

113 
114 


971  pounds. 

|-  of  a  cent  cost  ; 

I  of  a  cent  sold  for; 

-^^  gain  on  each  ; 

80  eggs  sold. 
84  years. 
942.'48+  cubic  feet. 


155.4.  -.iR.  38.72P. 

A,  25  days, 

B,  30  days, 

C,  37-^  days. 
$365,837  nearly 


118 
119 
120 
121 


(j-fjf  hours. 
lOSy"^^  planks. 
5  inches. 
■^4006,54  + 


3o2. 


2 
.■> 
o 


36  acres.  ! 

5 A  IR.   15P.     ! 


4   I  ]  35  acres. 


43G 


ANSWERS. 


P-  1 

EX. 

ANS. 

EX. 

5 
G 
7 
8 

ANS. 

353.! 
353.! 
353.' 
353. 

1 

2 
3 

4 

437 A   2R.  3lP-i- 
291^.  2/i.   IGP. 
35^.   Oil.  2oP. 

20  A 

40^. 

\bA. 

2\A,    ]R.  8  P. 

26  A.   3Ii.  20 P.  0,/d. 

354. 
354. 
354. 


o 

4 


21 A   OR.  39.824P. 
921.875.V.7./5. 
704.12o.v^.  V^. 


0 
6 


60  A  SR.    12. 8 P. 
270 A   IR.  24  P. 


355.1    2 
355.  i  3 


356. 


o»4.33/G 
125.664 


4 
2 


17  9.U712 
7418 


4360.8354- 


19.635+       I   3   I    153  9384     ||  4  |     1.069 +.S7.  yc?. 


357.1 
357.1 

358. 
358. 
358. 


359. 
359. 


615.7536 
4071.5136 


19699657 1.7221046^.  mi 


2 

»■> 
o 

4 


268.0832 

2144.6656  c.  in. 

2599927920826.6374908 


0 

1 
2 


904.7808  c./(5. 

giOOsy./^. 

]440.s^y./i;. 


2 
3 


110592  c.  f/z. 
42|c.//. 


4 
5 


315f^  gallons. 
13820  eft. 


360. 

2 

360. 

0 

360. 

4 

360. 

2 

361. 
361. 


233.334- 


^  sq.ft. 
2827.44  sy.  zn. 
6283.2s(7./;;. 
36442.56 


4 
5 


13571  712 

9650.9952 
7363.125 


2 

o 
O 


4380 
2484 


5620 
5760 


6 

7 


14400 
1800 


362.  i  2      9160.9056 


8659.035 


4  1   2827.44 


364.1 

2 

32.4938i/z.     |l  3      28.2574i;z.                      

365. 
365. 
365. 

1 
2 

197.459-1-   gal.  wine. 
162.613+   gal.  beer. 

3 

4 

(  136.9209+  wine  gal. 

(112.7583+  beer  gal. 

148.3772+  wine  gal. 

367.11    1   I  40ZA. 

36S.I    5 
368.!    6 


25/6. 


50/*. 


II   4   I   2U/A. 


AOlb. 

l/ji.,  lirin.,  2m.,  Ain. 


7 
8 


64//;. 
150/6. 


370.11   1   I   60//?. 


2  I   40/6. 


II  3  I   25/6. 


371.11   1   1  Ufi.                li  2  1   U//. 

372.11   1  1  40/6.      11  2  1   100/6.     ||  3 

j   60/6.     II    1 

576/6. 

373.11  2  1  2250/6.               — 

374.11  1  1  259200/6.11  2  |1.47  +  /6.1l  3  I 

1.1  +  /6.II  4  1 

1.2/??. 

ANSWEKS. 


437 


p. 

KX. 

ANS. 

EX. 

ANS. 

375. 

1 

23»z/.  2760/^. 

7 

Gda.  iWu: 

375. 

2 

57  GO/"/-. 

8 

131 A 

875. 

3 

21ir.  56  m. 

9 

Gini.  050 5. ^  ft. 

375. 

4 

8fi. 

10 

Shr.  \2>n.  12||.5ec. 

375. 

5 

2'3^sec. 

HA 

11 

8m.  IG.Gsrc. 

375. 

6 

12 

1  G4.285?m. 

377. 

1 

377. 

377. 

o 

377. 

^.< 

377. 

o 

o 

377. 

4 

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1447.5//. 
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..^319 

1  wa 

4    S.   BARNES  &  COMPANy's  PUBLICATIONS. 
Page's  Thtory  and  Practice  of  Teaching. 


THEORY  AND  PRACTICE  OF  TEACHING? 


MOTIVES    OF    GOOD    SCHOOL-KEEPING. 

BY  DAVID  PAGE,  A.M., 

LATB  PRINCIPAL  Or  THE  STATE  NORMAL  SCHOOL,  SKW  YORK. 


•I  received  a  few  days  since  your  'Tlioory  and  Practice,  &c.,'  and  a  capital  (A«*r» 
Bud  capital  /iractir.e  it  is.  I  liMve  i\-:u\  it  with  iiiiiiuiiv'lfcl  ilclii;lit.  Ev<ii  il'  I  Hhctlld 
look  Ihi-oiigli  a  critic's  microscope,  I  should  iKinlly  tiiul  a  siii'^xle  seiiliiiii  iil  to  (li.»«)nj 
from,  and  cerlaiiilv  not  one  to  coikIuiiiii.  The  cliapters  on  I'rizes  am!  on  Ctn;i(iTal 
Puiii.</iiiir-nt  are  tniiy  adiiiiralile.  Tlicy  will  i^xcrt  a  niosi  salutary  iiiflui-ncc.  So  .,f  tho 
news  s/uir.-iiiii  oil  moral  ai;;!  rt'li^nous  iiislruclioii,  wliich  you  so  fanicsily  niiil  IV-lin^ly 
liitiisl  upon,  and  yi-l  wiilun  true  Protfstanl  limits.  It  is  a  okand  book,  A.sri  I  ti;ank 
Ukavkn  that  yi)I'  uav^  written  it." — Jluu.  Hurace  JSlann,  istcrUary  uf  Ihe  liuard  uj 
Eduiuittva  tn  Mamackiisctts. 


"•Were  it  our  businoss  to  examine  teachers  we  would  never  dismi?s  a  candidal* 
without  namiii','  this  hook.  Other  things  bcui;,'  eipial,  we  woulil  iricatly  prf  "iT  a  leachor 
who  has  reail  it  and  speaks  of  it  wilh  enthusiasm.  In  one  in(liireiei.l  to  such  a  work, 
we  should  cerlainly  have  little  conlideiice,  however  he  mii,'ht  appear  in  other  respecla 
Would  that  ever)  "teacher  employed  in  Vermont  this  winter  had  'he  spirit  of  this  bvjk 
In  Uib  bosom,  itslessoiis  impressed  upon  his  hviuiV—  h'crmuiil.  Lhroiucle. 


"1  nm  plea.«ed  with  and  commenil  this  work  to  the  attention  of  scIkhiI  teachers,  and 
those  who  intend  to  embrace  th.it  most  eslunabie  prolession,  tor  li'.;hl  and  instruction 
to  g'.lide  and  jfovern  them  in  the  discliaixe  of  their  dehcale  and  imporUul  dulieS."— 
AT.  S.  liciUuii,  HuyeruileiuLeiil  uf  Cumiiwn  fic/tuols,  State  of  A'cw  i'ur.'c. 


Bon.  S.  Young  says,  "  It  is  aliogether  the  best  book  on  this  subject  1  have  evei 
seen." 


Frfsident  .Vurt!,,  of  HamiJtnn  Col/esre,  says,  "1  have  read  it  with  al!  that  absorbing 
self-deiiyins;  inu-rest,  which  in  my  younger  days  waa  reserved  for  ticlion  mid  poetry.  I 
am  delighted  with  the  book." 

ifnn.  Marcus  S.  fininnJds  says,  "  It  will  do  pcreat  good  by  showinc;  the  Tcncher  what 
ibould  be  his  qualificJlious,  and  what  may  justly  be  requiicd  and  expected  of  him." 


"1  wish  you  would  send  an  asent  tlirough  the  several  towns  of  lliis  Stcte  with 
Pages  'Theijry  and  Practice  of  Teachinu','  or  take  some  other  way  of  Ijrin^ing  this 
falisab'.o  book  to  the  nolice  of  every  family  and  u\  every  l';aclier.  1  should  be  rejoiced 
to  see  the  principles  which  it  presents  as  to  the  motivi-s  ami  methods  of  i;oud  ?chc«>}- 
keepiii?  carried  ul  in  every  school-room  :  and  as  nearly  as  possil)liN  in  llie  style  ia 
wh':3h  Mr.  Pa^e  illustrates  them  in  his  own  practice,  as  the  devoted  and  acconiplifhud 
Principal  of  your  t^tate  Normal  School."— //c;n-y  Barnard^  SupcnnUndciil  vf  Comnuv 
Sthooii  for  the  atate  uf  Rhode  Island. 

"The  'Theory  and  Practice  of  Teachins,'  by  D.  P.  Page,  is  one  of  Ihe  best  ()ookB  oJ 
the  kind  1  have  ever  met  with.  In  it  the  tlieory  and  practice  of  the  teacher's  duties 
ere  clearly  explained  and  happilv  combined.  The  style  is  easy  and  fainihar,  and  the 
BUgKcstious  it  contains  are  plain,  jiraclical,  and  to  the  point.  To  leachers  especial!?  d 
wUl  furnish  very  imporUmt  aid  in  dischiu-sing  the  diitie«,of  Jieir  hc;h  and  respuasiiila 
profeMion."— /eoger  6'.  Howard^  Siuieriiitendent  of  Ck'mnion  SchuoU,  Uratss  I'o.y  t'U 


A.    8.    BARNES    AND    COMPANV's    PUBLICATIONS. 
No rthend' s    Teacher    and  P arenl, 

A    NEW    VOLUME    FOR    THE    TEACHEr's    LTBRAET. 

THE  TEACHER  AND  THE  PARENT: 

A.  Treatise  upon  Common-Scliool  Education,  containing  Practical  Sug- 
gestions to  Teachers  and  Parents.  By  Charles  Northend,  A.  M, 
late,  and  for  many  years,  Principal  of  the  Epes  Scliool,  Salens.  Now 
Su'Derintendent  of  Public  Schools,  Danvers,  Mass. 


■■•"Wfi  may  anticipate  for  tbis  work  a  -wide  circulation,  among  teachers  and  friends 
of  education.  The  extensive  and  liigb  reputation  of  its  autlior,  indeed,  will  bespeak 
for  it  more  than  pen  of  ours  can  do.  It  is  a  work  of  about  three  hundred  and 
twenty  pages,  in  good  size  tj'pe,  and  presents  a  very  plea.'^ant  appearance  to  the  eyc^ 
BS  well  as  the  work  noticed  on  the  preceding  page,  both  of  which,  for  their  neat 
appearance,  do  great  credit  to  the  enterprising  publishers. 

Mr.  Northend's  book  will  prove  interesting  to  all,  and  of  great  benefit  to  teach- 
ers, especially  as  a  chart  for  those  just  commencing  to  engage  in  the  profession. 
As  a  r.aOe  mecum,  it  will  prove  a  very  ple.asant  companion,  lor  its  pagi-s  are  tilled 
with  the  results  of  a  large  experience  presented  in  a  very  pleasing  form.  We  are 
glad  to  find  that  the  author,  in  furnishing  to  teachers  so  useful  a  work,  has  not 
neglected  the  svaviter  in  modo,  and  has  here  and  there  thrown  in  a  pleasant  anec- 
dote, which  will  enliven  its  character,  and  make  it  all  the  more  acceptable.  Wo 
shall  have  frequent  occasion  to  refer  to  it  hereafter.  In  closing  this  short  notice, 
we  would  assure  our  readers  that  a  perusal  of  the  work  will  more  than  realize  to 
them  the  truth  of  all  we  have  attempted  to  say  in  its  favor.  Appended  to  tho 
volume  will  be  found  a  catalogue  of  educational  works  suitable  for  tho  teachers 
library." — Massachusetts  Teacher. 


"We  wish  that  this  interesting  and  read.ablo  volume  may  find  a  place  in  every 
family,  and  we  are  certain  that  it  ought  to  be  on  the  shelf  of  every  school  library  in 
the  land."— 5aZem  Gazette. 


"It  presents  a  multitude  of  practical  hints,  which  cannot  fail  to  do  good  service  In 
enlightening  all  laborers  in  the  tield  of  education." — £ostvn  Transcript. 


"We  unhesitatingly  commend  this  volume  of  sound,  practical,  common  sense  snj- 
pestions.  Every  seboid  teacher  should  carefully  examine  its  paces,  and  ho  will  not 
fail— he  cannot  help  receiving — invaluable  aid  therefrom." — Boxton  Atlas. 


"We  h.ivo  examined  this  work  with  care,  and  cheerfully  commend  it  t?  parenti 
Md  teachers.  It  abounds  in  judicious  advice  and  sound  reasoning,  and  cannot  fail  to 
Impart  ideas  in  the  education  of  children  which  may  be  acted  upon  with  the  most 
beneficial  results." — Boston  Mercantile  Journal. 


""Kiis  is  an  Intelligible,  practical,  and  most  excellent  treatise.  The  book  is 
•nlivened  with  numerous  anecdotes  which  serve  to  clinch  the  sood  advice  given,  m 
well  as  to  keeji  awake  iho  attention  of  the  advised." — Boston  Traveller. 


"Tub  la  a  sterling  work  of  groat  value.    It  should  be  In  every  fimlljr.    A!^  t««oh 
ITS  need  just  «uch  a  work." — Boston  Olive  Branch. 


A.  8.  BARNES  A  COMPANY'S  PUBI.ICATT0N8. 


Mansfield    on    American    E  du  ea  tion. 


/MERICAN      education; 

ITS    PRINCIPLES    AND    ELEIIENTS. 

DEDICATED    TO    THE    TEACHERS    OF    THE    UNITED    ST/'l'I.i 

BY  EDWARD  D.  MANSFIELD, 
Author  of  ^'•I'olitical  G-rammar,''''  etc. 

This  work  is  suggestive  of  principles,  and  not  intended  to  point  o;-*  r, 
eourse  of  studies.  Its  aim  is  to  excite  attention  to  what  sliould  be  tii« 
elements  of  an  American  education ;  or,  in  other  words,  what  are  thfl 
ideas  connected  with  a  republican  and  Christian  education  in  this  period 
of  rapid  development. 

"The  author  could  not  have  applied  his  pen  to  the  production  of  a  book  upon  a 
subject  of  more  importance  th:m  the  one  he  has  chosen.  We  have  had  occasion  to 
notice  one  or  two  new  works  on  education  recently,  whi^h  indicate  tliat  the  attention 
of  authors  is  beini;  directed  toward  that  suljject.  We  trust  that  those  who  occupy  tho 
proud  position  of  teachers  of  American  yoiUli  will  find  much  in  these  worlis,  wliich  are 
a  sort  of  inlerchanc;e  of  opinion,  to  assist  them  in  tlie  discharge  of  their  responsibie  d'.ities, 

"The  author  of  the  work  lietbre  us  does  not  point  out  any  particular  course  of  sliulies 
to  be  pursued,  but  conliues  hirasell  to  the  consideration  of  the  principles  which  should 
govern  teachers.  His  views  upon  the  elements  of  an  American  education,  and  ila 
bearings  upon  our  institutions,  are  sound,  and  worthy  the  attention  of  those  to  whom 
they  are  paiticularly  addressed.  We  commend  the  work  to  teachers." — Jiuohester 
Daily  Advertiser, 

"We  have  examined  it  with  some  care,  and  are  delighted  with  it.  It  discusses  the 
whole  subject  of  American  education,  and  presents  views  at  once  enlarged  and  compr& 
hensive ;  it,  in  fact,  covers  the  whole  ground.  It  is  high-toned  in  its  moral  ana 
religious  bearing,  and  points  out  to  the  student  the  way  in  which  to  be  a  man.  It 
ehoidd  be  in  every  public  and  private  library  in  the  country." — Jackson  Patrwt. 


"  It  is  an  elevated,  dignified  work  of  a  philosopher,  who  has  written  a  book  on  tho 
Bubject  of  education,  which  is  an  acquisition  of  great  value  to  all  classes  of  our 
Countrymen.  It  can  be  read  with,  interest  and  profit,  by  the  old  and  young,  the 
educaled  and  unlearned.  We  hail  it  in  this  era  of  superficial  and  ephemeral  litera- 
ture, iLM  the  precursor  of  a  better  future.  It  discusses  a  momentous  subject;  bringing 
to  bear,  in  its  examination,  the  deep  and  labored  thought  of  a  comprehensive  mind. 
We  hope  its  sentiments  may  be  diffused  as  freely  and  as  widely  throughout  our  land 
88  the  air  we  breathe." — Kalamazoo  Oazctte, 


"  Important  and  comprehensive  as  is  the  title  of  this  work,  we  assure  our  readers  11 
[fi  no  misnomer.  A  wide  gap  in  the  bulwark  of  lliis  age  and  this  country  is  greatly 
lossenetl  by  this  excellent  book.  In  the  first  place,  the  views  of  the  author  on  educa- 
tion, irrespective  of  time  and  place,  are  of  the  highest  order,  contrasting  strongly  with 
too  groveling,  time-seeking  views  so  plausible  and  so  popular  at  the  present  day. 
A  leading  purpose  of  the  author  is,  as  he  says  in  the  preface, '  to  turn  the  thoughts  ol 
those  engaged  in  the  direction  of  youth  to  the  fact,  thai  it  is  the  entire  soul,  in  all  ita 
fgiculties,  which  needs  education.' 

'•The  views  of  the  author  are  eminently  philosophical,  and  he  does  not  pretend  to 
enter  into  the  details  of  teaching:  but  his  is  a  practical  philosophy.  Iiaviug  to  do  with 
Ufine,  abiding  truths,  and  does  not  sneer  at  utility,  though  it  demands  a  utility  thai 
takes  hold  of  the  spiritual  part  of  man,  and  reaches  into  bis  immortality." — Uolden'i 


A.  S,  BARNES   k  COMPANY'S  PUBLICATICN8. 
J)  e  Tocquevill  c'  s   American    Ins  t  itutione. 

AMERICAN     INSTITUTIONS     AND    THEIR    INFLUENCE. 

BY  ALEXIS  DE  TOCQUEVILLE. 

WITH  NOTES,  BY  HON.  JOHN  C.  SPENCER. 1  vol.  8vo. 

This  book  is  the  first  part  of  De  Tocqueville's  larger  work,  on  the  Repiblio  ot 
America,  and  is  one  of  llie  most  valuable  treatises  on  American  poiiiics  that  has  eTM 
t>i»n  issued,  and  should  be  in  every  library  in  the  land.  The  views  of  a  Hbarai- 
miaded  and  enlightened  European  statesman  upon  the  working  of  our  coiuitry's  socia) 
said  political  establishments,  are  worthy  of  attentive  perusal  at  all  times;  those  of  a  raa» 
lite  Ue  Tucqueville  have  a  higher  iuti'insic  value,  from  the  fact  of  his  residence  among 
the  people  he  describes,  and  his  after  position  as  a  part  of  the  republican  government 
of  France.  Tlie  work  is  enriched  likewise  with  a  preface,  and  carefully  prepared  notes, 
by  a  well-known  American  statesman  and  late  Secretary  of  the  Navy.  The  book  is  on« 
of  great  weight  and  inleresi,  and  is  admirably  adapted  for  the  district  and  school  library 
as  well  as  that  of  the  private  student.  It  traces  the  origin  of  the -Anglo-American* 
treats  of  their  social  condition,  its  essential  democracy  and  [lolitical  consequences,  tb# 
•overeignty  of  the  people,  etc.  It  also  embraces  the  author's  views  on  the  .Ameriaii 
system  of  townships,  counties,  &c. ;  federal  and  state  powers;  the  judiciary  ;  llie  cod 
Blitution  ;  parties;  the  press ;  American  sociely  ;  jjower  of  the  mijority,  its  tyrann\ 
and  the  causes  which  mitigate  it ;  trial  by  jury;  religion;  the  three  races;  the  arista 
cratic  party ;  causes  of  American  commercial  prosperity,  etc.,  etc.  The  work  is  ai 
epitome  uf  the  entire  political  and  social  condition  of  the  United  Slates. 

"M.  De  Tocqueville  was  the  first  foreign  author  who  comprehended  the  genius  c» 
our  institutions,  and  who  made  intelligible  to  Europeans  the  complicated  ra.ichliierj- 
wheel  within  wheel,  of  the  stale  and  federal  governments.  His  '  Democracv  ic 
America'  is  ackiiowled'-'ed  to  be  the  most  profound  and  philosophical  work  iipor 
rnodern  reiiublieanlsm  that  has  yet  appeared.  It  is  characterized  by  a  rare  uniiiii  o 
discenimenl,  rellecliiin,  and  candor;  and  though  occasionally  tinged  with  the  aiilhor'f 
pecuiiarilies  of  education  and  faith,  it  may  he  accepted  as  In  the  main  a  just  and  iu> 
partial  criticism  upon  the  social  and  political  feiitures  ol  the  United  Slates.  The  pui> 
lishers  have  now  sought  to  adapt  it  as  a  te.\t-book  for  higher  seminaries  of  learninji 
Fi'r  this  purpose  they  have  published  the  lirsl  volume  as  an  independent  work,  ihut 
avoiding  the  auihur's  speinilalions  ujiDn  our  social  habits  and  reli'-rioiis  cnmlitinii.  Thi» 
volume,  however,  is  unmiililaled — I  he  author  ia  left  throu^'hout  to  speak  for  himself;  bill 
where  at  any  pnjnl  he  had  misappiehended  our  sjstem.  the  detect  is  sujiplied  by  iiule! 
or  para:,'raphs  in  brackets  from  the  pen  of  one  inosi  thoroughly  versed  in  the  hislnri 
the  legislalioii,  the  adininislration,  and  the  jurisprudence  of  our  counlry.  This  woi^k 
will  supply  a  felt  deliciency  in  the  educational  apparatus  of  our  higher  schools.  Kveri 
man  wlio  pretends  lo  a  good,  and  much  more  to  a  liberal  ediic;ition,  should  ma.-;i  i  th'f 
5)rinciples  ami  philosophy  of  the  inslitutionsorbis  coiintiy.  In  the  hands  of  ajuu.ciou* 
teacher,  Hits  volume  will  be  an  adniir.ible  te\t-bi)ok." — The  huUjicndcnl. 

'■'  Having  had  the  honor  of  a  personal  acqiiaiiitance  with  .M.  De  Tocqueville  while  h# 
was  in  this  country ;  having  discussed  with  him  many  of  the  topics  treated  of  in  Ibb 
bo<jk  ;  having  entered  deeply  into  the  feelings  and  sentiments  which  guided  and  iu> 
pelled  him  In  his  Uisk,  and  having  formed  a  hi;;h  admiration  of  his  chaia-uer  and  rf 
this  production,  the  editor  felt  under  some  obligation  lo  aid  in  procuring'  for  one  whoix 
he  ventures  to  call  his  friend,  a  hearini;  from  Ihose  who  were  Ihe  olijecl-s  of  his  ob' 
»ervalions.'  The  notes  of  Mr.  Spencer  will  be  found  to  elucidate  occasional  niiacc* 
ceplions  of  the  transl.iior.  It  is  a  most  judicious  text-book,  and  ought  lo  be  reivC 
wrefully  by  all  who  wish  to  know  this  country,  and  to  trace  its  power,  position,  an^ 
nllimate  deslit/y  from  the  true  source  of  philosophic  governmi'iii,  Ivejuiblicanism  — th« 
people.  De  Toc<iueville,  believing  the  destinies  olcivilizalion  lo  depend  on  the  p<.wer 
of  the  jx'ople  and  on  the  principle  which  so  ^'randly  founded  iin  exponent  on  this  roB 
linenl,  analyzes  with  jealous  care  and  peculiar  critical  acumen  the  tendencies  of  lh« 
DOW  Democracy,  and  candidly  gives  his  approval  of  the  new-born  giaiil,  or  pointt 
out  ami  warns  him  of  daimers  which  his  faillifiil  and  independent  philosophy  foreseen 
We  birlieve  the  perusal  ot  his  observations  will  have  Iho  ell'ecl  of  enhancing  still  nior* 
lo  •■  is  Ainerii:an  readers  the  stiiiclure  ol  their  governnifml,  by  the  clejir  and  proloiin/ 
W^-'e  in  which  !.«*  prfajtiitsi  U.^^./'/mcncun  Jircteu. 


Davies'  System  of  Mathematics. 


DAVIES'    LOGIC  OF  MATHEMATICS. 
Tlie  Logic  and  Utility  of  Mathematics,  with  the  best  imthods  of  Inatruo 
tion,  explained  and  illustrated.     By  Chaules  Davies,  L.  L.  D. 

"One  of  the  most  remarkable  books  of  the  month,  is  '  The  Lo;,'ic  and  Utility  of 
Malhemiitics,  by  Cb;irles  Davies,  1,.  L.  1).,'  pnlili'^lied  by  Barnes  &.  Co.  ll  is  not  iu- 
tendi'd  as  a  treatise  on  any  special  br.incli  of  niailieinatical  science,  ami  demani's  fm 
its  full  a|i|>recialiun  a  general  actjiiainlance  with  the  leading  iiictbods  ami  routine  of 
niatliemalical  invesiiijation.  'I'o  tho  e  who  have  a  natural  fondness  lor  this  (iiirsuit 
and  enjoy  the  leisure  for  a  retrospect  of  their  fivorite  studies,  the  pre.sent  \olunie  wil 
possess  a  charm,  not  surpassed  by  tiie  (asciiiations  of  a  romance,  it  is  an  efiliorale 
and  Iticid  exposition  of  the  principles  which  lie  at  the  foundation  of  pure  malliematicu 
with  a  hit;hly  inuenions  application  of  their  results  to  the  development  of  the  essen- 
tial idea  id'  .Arilhnietic.  (Jeinoelry,  Al'.'i'bra,  .-Xnalytic  (Jeometry,  and  the  DuiiTi-ntial 
and  Inteural  Calculus.  The  work 'is  preceded  by  a  ;,'eneral  view  of  the  subject  of  Lo;;ic, 
mainly  drawn  from  the  writinj:s  of  Arclibisbop  Whately  and  Mr.  Mill,  and  closes  «ith 
an  essiiy  on  the  utility  of  mathematics.  Some  occasional  e.xa^'geralions,  in  presenting 
the  clai'ms  of  the  science  to  which  his  life  has  been  devoted,  UiUst  here  be  pardoned 
to  the  professional  enthusiasm  of  the  author.  In  general,  the  work  is  wriitea  with 
s'.ngnlar  circuniS(jection  ;  the  views  of  the  best  thinkers  on  the  subject  have  been 
thorouuhly  digcaled,  and  are  presented  in  an  orij.'inal  form;  every  thin<;  bears  the  im- 
press of  llie  intellect  of  the  writer  ;  his  style  is  for  the  most  part  chaste,  simple,  trans- 
parent, and  in  admirable  harmony  with  the  dif;nity  of  the  subject,  and  his  condensed 
generalizations  are  often  profound  ai.d  always  suj^gestive."— i/ar//er's  A'cic  jUu«£Aij 
Maj;aiinc. 

"This  work  is  not  merely  a  mathematical  treatise  to  be  used  as  a  te.\t  book,  but  a 
complete  and  philosophical  unfolding  of  the  principles  and  truths  of  luathematical 
science. 

"  ll  is  not  only  designed  for  professional  teachers,  professional  men,  and  students  ol 
mathematics  and  philosophy,  but  for  the  general  reader  who  desires  mental  improve- 
meni.  and  would  learn  to  search  out  the  miport  of  language,  and  ac(iuire  a  habit  of 
notiiit;  of  conni'xion  between  ideas  and  their  signs;  also,  of  the  relation  of  ideas  to 
each  other. —  The  Student. 


"  Students  of  the  Science  will  find  this  volume  full  of  useful  and  deeply  interesting 
matter." — .Itbany  Evening  Juuruixl. 


"  Seldom  have  we  opened  a  book  so  attractive  .is  this  in  its  typography  and  style  ol 
eiccution  ;  and  there  is  besides,  on  the  margui  opposite  each  section,  an  index  of  the 
subject  (d"  whicli  it  treats— a  great  convenience  to  the  student.  I!ut  the  mutter  is  no 
less  to  be  commended  than  the  manner.  And  we  are  very  much  mi-taken  if  ibis  wurk 
sliall  not  prove  mnre  popular  and  more  useful  than  any  which  the  distinguished  author 
has  given  to  tlie  public."' — Lutheran  Observer. 


"  We  have  been  much  interested  both  in  the  plan  and  in  the  execution  of  the  work. 
Iinil  would  recommend  the  study  of  it  to  ilie  tlieologian  as  a  discipline  in  close  ami 
Kcurate  tliinking.  aMi  in  logical  nielhod  ar.d  reasoning.  It  will  be  useful,  al-o,  to  thf 
general  scholar  and  to  tlie  practical  muchanic.  We  w(Uild  s|iecially  rcconiiiiend  it  M 
those  who  would  have  nothing  tauglit  in  our  Free  .Acailemy  and  other  liiglier  inslitu 
Uons  but  what  is  directly  'practical';  nowhere  have  wa  seen  a  finer  iiluslraliun  o< 
the  eoiuieetion  between  the  ab.straetly  scieiitific  and  tlie  praclic.-il. 

"The  work  is  divided  into  three  books;  llie  first  of  wliich  treats  of  Logic,  mainly 
ipon  tlie  liasis  of  Whately;  the  second,  of  Matlieuiatical  Science;  aud  Iho  third,  r  f  the 
Dtility  of  },liilhemixtic3."—Incl^2^endent. 

"Tlie  authoi's  style  is  perspicuous  and  concise,  and  he  exliibits  a  in.istery  of  the 
ibstni.se  topics  which  he  attempts  to  .simplify.  For  the  inatbeiiiatical  student,  ^^■!l^ 
desires  an  analytical  kiunvled^'e  of  the  science,  and  who  would  begin  at  the  beginning. 
we  .should  suppose  the  work  would  have  a  special  utility.  I'rof.  Davies"  matbemati- 
cal  works,  we  liolieve,  have  become  quite  popular  with  educators,  and  this  di.selo.soj 
qidte  .IS  much  reasearch  aud  practical  scliolarship  a.s  any  we  havp  seen  from  Ids  p^u  ' 

•A>i(i-  Yi  }'k  EvangelUst 


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